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Theorem iunrelexp0 38496
Description: Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
Assertion
Ref Expression
iunrelexp0 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
Distinct variable groups:   𝑥,𝑅   𝑥,𝑉   𝑥,𝑍

Proof of Theorem iunrelexp0
StepHypRef Expression
1 df-pr 4324 . . . . . . 7 {0, 1} = ({0} ∪ {1})
21ineq1i 3953 . . . . . 6 ({0, 1} ∩ 𝑍) = (({0} ∪ {1}) ∩ 𝑍)
3 indir 4018 . . . . . 6 (({0} ∪ {1}) ∩ 𝑍) = (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))
42, 3eqtr2i 2783 . . . . 5 (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) = ({0, 1} ∩ 𝑍)
54uneq1i 3906 . . . 4 ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) = (({0, 1} ∩ 𝑍) ∪ 𝑍)
6 inss2 3977 . . . . 5 ({0, 1} ∩ 𝑍) ⊆ 𝑍
7 ssequn1 3926 . . . . 5 (({0, 1} ∩ 𝑍) ⊆ 𝑍 ↔ (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍)
86, 7mpbi 220 . . . 4 (({0, 1} ∩ 𝑍) ∪ 𝑍) = 𝑍
95, 8eqtr2i 2783 . . 3 𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)
10 iuneq1 4686 . . . 4 (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → 𝑥𝑍 (𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥))
1110oveq1d 6828 . . 3 (𝑍 = ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0))
129, 11ax-mp 5 . 2 ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0)
13 dmiun 5488 . . . . . . 7 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥)
14 iunxun 4757 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)dom (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
15 iunxun 4757 . . . . . . . . . 10 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
1615equncomi 3902 . . . . . . . . 9 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) = ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))
1716uneq1i 3906 . . . . . . . 8 ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) = (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
1817equncomi 3902 . . . . . . 7 ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) = ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)))
1913, 14, 183eqtri 2786 . . . . . 6 dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)))
20 rniun 5701 . . . . . . 7 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥)
21 iunxun 4757 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)ran (𝑅𝑟𝑥) = ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
22 iunxun 4757 . . . . . . . 8 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) = ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))
2322uneq1i 3906 . . . . . . 7 ( 𝑥 ∈ (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍))ran (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) = (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
2420, 21, 233eqtri 2786 . . . . . 6 ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) = (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))
2519, 24uneq12i 3908 . . . . 5 (dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
26 uncom 3900 . . . . . . 7 ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥))
2726uneq1i 3906 . . . . . 6 (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
28 un4 3916 . . . . . 6 ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ 𝑥𝑍 dom (𝑅𝑟𝑥)) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
2927, 28eqtri 2782 . . . . 5 (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥))) ∪ (( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
30 uncom 3900 . . . . . . . 8 ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) = ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥))
3130uneq1i 3906 . . . . . . 7 (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)))
32 un4 3916 . . . . . . 7 (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)))
3331, 32eqtri 2782 . . . . . 6 (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)))
3433uneq1i 3906 . . . . 5 ((( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
3525, 29, 343eqtri 2786 . . . 4 (dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)))
36 df-ne 2933 . . . . . . . . . 10 (({0, 1} ∩ 𝑍) ≠ ∅ ↔ ¬ ({0, 1} ∩ 𝑍) = ∅)
37 incom 3948 . . . . . . . . . . . . . . 15 ({0, 1} ∩ 𝑍) = (𝑍 ∩ {0, 1})
381ineq2i 3954 . . . . . . . . . . . . . . 15 (𝑍 ∩ {0, 1}) = (𝑍 ∩ ({0} ∪ {1}))
39 indi 4016 . . . . . . . . . . . . . . 15 (𝑍 ∩ ({0} ∪ {1})) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4037, 38, 393eqtri 2786 . . . . . . . . . . . . . 14 ({0, 1} ∩ 𝑍) = ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1}))
4140eqeq1i 2765 . . . . . . . . . . . . 13 (({0, 1} ∩ 𝑍) = ∅ ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
42 un00 4154 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ((𝑍 ∩ {0}) ∪ (𝑍 ∩ {1})) = ∅)
43 anor 511 . . . . . . . . . . . . 13 (((𝑍 ∩ {0}) = ∅ ∧ (𝑍 ∩ {1}) = ∅) ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4441, 42, 433bitr2i 288 . . . . . . . . . . . 12 (({0, 1} ∩ 𝑍) = ∅ ↔ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
4544notbii 309 . . . . . . . . . . 11 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
46 notnotb 304 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ ¬ ¬ (¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅))
47 disjsn 4390 . . . . . . . . . . . . . 14 ((𝑍 ∩ {0}) = ∅ ↔ ¬ 0 ∈ 𝑍)
4847notbii 309 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {0}) = ∅ ↔ ¬ ¬ 0 ∈ 𝑍)
49 notnotb 304 . . . . . . . . . . . . 13 (0 ∈ 𝑍 ↔ ¬ ¬ 0 ∈ 𝑍)
5048, 49bitr4i 267 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {0}) = ∅ ↔ 0 ∈ 𝑍)
51 disjsn 4390 . . . . . . . . . . . . . 14 ((𝑍 ∩ {1}) = ∅ ↔ ¬ 1 ∈ 𝑍)
5251notbii 309 . . . . . . . . . . . . 13 (¬ (𝑍 ∩ {1}) = ∅ ↔ ¬ ¬ 1 ∈ 𝑍)
53 notnotb 304 . . . . . . . . . . . . 13 (1 ∈ 𝑍 ↔ ¬ ¬ 1 ∈ 𝑍)
5452, 53bitr4i 267 . . . . . . . . . . . 12 (¬ (𝑍 ∩ {1}) = ∅ ↔ 1 ∈ 𝑍)
5550, 54orbi12i 544 . . . . . . . . . . 11 ((¬ (𝑍 ∩ {0}) = ∅ ∨ ¬ (𝑍 ∩ {1}) = ∅) ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5645, 46, 553bitr2i 288 . . . . . . . . . 10 (¬ ({0, 1} ∩ 𝑍) = ∅ ↔ (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
5736, 56sylbb 209 . . . . . . . . 9 (({0, 1} ∩ 𝑍) ≠ ∅ → (0 ∈ 𝑍 ∨ 1 ∈ 𝑍))
58 simpl 474 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 0 ∈ 𝑍)
5958snssd 4485 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {0} ⊆ 𝑍)
60 df-ss 3729 . . . . . . . . . . . . . . . . . . 19 ({0} ⊆ 𝑍 ↔ ({0} ∩ 𝑍) = {0})
6159, 60sylib 208 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) = {0})
6261iuneq1d 4697 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {0}dom (𝑅𝑟𝑥))
63 c0ex 10226 . . . . . . . . . . . . . . . . . 18 0 ∈ V
64 oveq2 6821 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 0 → (𝑅𝑟𝑥) = (𝑅𝑟0))
6564dmeqd 5481 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
6663, 65iunxsn 4755 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}dom (𝑅𝑟𝑥) = dom (𝑅𝑟0)
6762, 66syl6eq 2810 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟0))
68 relexp0g 13961 . . . . . . . . . . . . . . . . . . 19 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6968ad2antll 767 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7069dmeqd 5481 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
71 dmresi 5615 . . . . . . . . . . . . . . . . 17 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
7270, 71syl6eq 2810 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
7367, 72eqtrd 2794 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
7461iuneq1d 4697 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {0}ran (𝑅𝑟𝑥))
7564rneqd 5508 . . . . . . . . . . . . . . . . . 18 (𝑥 = 0 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7663, 75iunxsn 4755 . . . . . . . . . . . . . . . . 17 𝑥 ∈ {0}ran (𝑅𝑟𝑥) = ran (𝑅𝑟0)
7774, 76syl6eq 2810 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟0))
7869rneqd 5508 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
79 rnresi 5637 . . . . . . . . . . . . . . . . 17 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8078, 79syl6eq 2810 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟0) = (dom 𝑅 ∪ ran 𝑅))
8177, 80eqtrd 2794 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) = (dom 𝑅 ∪ ran 𝑅))
8273, 81uneq12d 3911 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) = ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)))
83 unidm 3899 . . . . . . . . . . . . . 14 ((dom 𝑅 ∪ ran 𝑅) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
8482, 83syl6eq 2810 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
8584uneq1d 3909 . . . . . . . . . . . 12 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))))
86 relexpdmg 13981 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8786expcom 450 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
8887ralrimiv 3103 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
8988ad2antll 767 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
90 olc 398 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
9190ad2antrl 766 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ⊆ ℕ0𝑍 ⊆ ℕ0))
92 inss 3985 . . . . . . . . . . . . . . . . . . . 20 (({1} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({1} ∩ 𝑍) ⊆ ℕ0)
9391, 92syl 17 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) ⊆ ℕ0)
9493sseld 3743 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑥 ∈ ({1} ∩ 𝑍) → 𝑥 ∈ ℕ0))
9594imim1d 82 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({1} ∩ 𝑍) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
9695ralimdv2 3099 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
9789, 96mpd 15 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
98 iunss 4713 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
9997, 98sylibr 224 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
100 relexprng 13985 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
101100expcom 450 . . . . . . . . . . . . . . . . . 18 (𝑅𝑉 → (𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
102101ralrimiv 3103 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
103102ad2antll 767 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
10494imim1d 82 . . . . . . . . . . . . . . . . 17 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({1} ∩ 𝑍) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
105104ralimdv2 3099 . . . . . . . . . . . . . . . 16 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
106103, 105mpd 15 . . . . . . . . . . . . . . 15 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
107 iunss 4713 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
108106, 107sylibr 224 . . . . . . . . . . . . . 14 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
10999, 108unssd 3932 . . . . . . . . . . . . 13 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
110 ssequn2 3929 . . . . . . . . . . . . 13 (( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
111109, 110sylib 208 . . . . . . . . . . . 12 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
11285, 111eqtrd 2794 . . . . . . . . . . 11 ((0 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
113112ex 449 . . . . . . . . . 10 (0 ∈ 𝑍 → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
114 simpl 474 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 1 ∈ 𝑍)
115114snssd 4485 . . . . . . . . . . . . . . . . . 18 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → {1} ⊆ 𝑍)
116 df-ss 3729 . . . . . . . . . . . . . . . . . 18 ({1} ⊆ 𝑍 ↔ ({1} ∩ 𝑍) = {1})
117115, 116sylib 208 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({1} ∩ 𝑍) = {1})
118117iuneq1d 4697 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = 𝑥 ∈ {1}dom (𝑅𝑟𝑥))
119 1ex 10227 . . . . . . . . . . . . . . . . 17 1 ∈ V
120 oveq2 6821 . . . . . . . . . . . . . . . . . 18 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
121120dmeqd 5481 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
122119, 121iunxsn 4755 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}dom (𝑅𝑟𝑥) = dom (𝑅𝑟1)
123118, 122syl6eq 2810 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom (𝑅𝑟1))
124 relexp1g 13965 . . . . . . . . . . . . . . . . 17 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
125124ad2antll 767 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑅𝑟1) = 𝑅)
126125dmeqd 5481 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → dom (𝑅𝑟1) = dom 𝑅)
127123, 126eqtrd 2794 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) = dom 𝑅)
128117iuneq1d 4697 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = 𝑥 ∈ {1}ran (𝑅𝑟𝑥))
129120rneqd 5508 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
130119, 129iunxsn 4755 . . . . . . . . . . . . . . . 16 𝑥 ∈ {1}ran (𝑅𝑟𝑥) = ran (𝑅𝑟1)
131128, 130syl6eq 2810 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran (𝑅𝑟1))
132125rneqd 5508 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ran (𝑅𝑟1) = ran 𝑅)
133131, 132eqtrd 2794 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥) = ran 𝑅)
134127, 133uneq12d 3911 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
135134uneq2d 3910 . . . . . . . . . . . 12 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)))
13688ad2antll 767 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
137 olc 398 . . . . . . . . . . . . . . . . . . . . 21 (𝑍 ⊆ ℕ0 → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
138137ad2antrl 766 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ⊆ ℕ0𝑍 ⊆ ℕ0))
139 inss 3985 . . . . . . . . . . . . . . . . . . . 20 (({0} ⊆ ℕ0𝑍 ⊆ ℕ0) → ({0} ∩ 𝑍) ⊆ ℕ0)
140138, 139syl 17 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ({0} ∩ 𝑍) ⊆ ℕ0)
141140sseld 3743 . . . . . . . . . . . . . . . . . 18 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (𝑥 ∈ ({0} ∩ 𝑍) → 𝑥 ∈ ℕ0))
142141imim1d 82 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({0} ∩ 𝑍) → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
143142ralimdv2 3099 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
144136, 143mpd 15 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
145 iunss 4713 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
146144, 145sylibr 224 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
147102ad2antll 767 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
148141imim1d 82 . . . . . . . . . . . . . . . . 17 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ((𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥 ∈ ({0} ∩ 𝑍) → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
149148ralimdv2 3099 . . . . . . . . . . . . . . . 16 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
150147, 149mpd 15 . . . . . . . . . . . . . . 15 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
151 iunss 4713 . . . . . . . . . . . . . . 15 ( 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
152150, 151sylibr 224 . . . . . . . . . . . . . 14 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
153146, 152unssd 3932 . . . . . . . . . . . . 13 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → ( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
154 ssequn1 3926 . . . . . . . . . . . . 13 (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅))
155153, 154sylib 208 . . . . . . . . . . . 12 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅))
156135, 155eqtrd 2794 . . . . . . . . . . 11 ((1 ∈ 𝑍 ∧ (𝑍 ⊆ ℕ0𝑅𝑉)) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
157156ex 449 . . . . . . . . . 10 (1 ∈ 𝑍 → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
158113, 157jaoi 393 . . . . . . . . 9 ((0 ∈ 𝑍 ∨ 1 ∈ 𝑍) → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
15957, 158syl 17 . . . . . . . 8 (({0, 1} ∩ 𝑍) ≠ ∅ → ((𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅)))
1601593impib 1109 . . . . . . 7 ((({0, 1} ∩ 𝑍) ≠ ∅ ∧ 𝑍 ⊆ ℕ0𝑅𝑉) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
1611603com13 1119 . . . . . 6 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
162161uneq1d 3909 . . . . 5 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))))
16388adantr 472 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
164 ssel 3738 . . . . . . . . . . . . 13 (𝑍 ⊆ ℕ0 → (𝑥𝑍𝑥 ∈ ℕ0))
165164adantl 473 . . . . . . . . . . . 12 ((𝑅𝑉𝑍 ⊆ ℕ0) → (𝑥𝑍𝑥 ∈ ℕ0))
166165imim1d 82 . . . . . . . . . . 11 ((𝑅𝑉𝑍 ⊆ ℕ0) → ((𝑥 ∈ ℕ0 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥𝑍 → dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
167166ralimdv2 3099 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → (∀𝑥 ∈ ℕ0 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
168163, 167mpd 15 . . . . . . . . 9 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
169 iunss 4713 . . . . . . . . 9 ( 𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
170168, 169sylibr 224 . . . . . . . 8 ((𝑅𝑉𝑍 ⊆ ℕ0) → 𝑥𝑍 dom (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
171102adantr 472 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
172165imim1d 82 . . . . . . . . . . 11 ((𝑅𝑉𝑍 ⊆ ℕ0) → ((𝑥 ∈ ℕ0 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑥𝑍 → ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))))
173172ralimdv2 3099 . . . . . . . . . 10 ((𝑅𝑉𝑍 ⊆ ℕ0) → (∀𝑥 ∈ ℕ0 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) → ∀𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅)))
174171, 173mpd 15 . . . . . . . . 9 ((𝑅𝑉𝑍 ⊆ ℕ0) → ∀𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
175 iunss 4713 . . . . . . . . 9 ( 𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ∀𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
176174, 175sylibr 224 . . . . . . . 8 ((𝑅𝑉𝑍 ⊆ ℕ0) → 𝑥𝑍 ran (𝑅𝑟𝑥) ⊆ (dom 𝑅 ∪ ran 𝑅))
177170, 176unssd 3932 . . . . . . 7 ((𝑅𝑉𝑍 ⊆ ℕ0) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
1781773adant3 1127 . . . . . 6 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅))
179 ssequn2 3929 . . . . . 6 (( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥)) ⊆ (dom 𝑅 ∪ ran 𝑅) ↔ ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
180178, 179sylib 208 . . . . 5 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((dom 𝑅 ∪ ran 𝑅) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
181162, 180eqtrd 2794 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((( 𝑥 ∈ ({0} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({0} ∩ 𝑍)ran (𝑅𝑟𝑥)) ∪ ( 𝑥 ∈ ({1} ∩ 𝑍)dom (𝑅𝑟𝑥) ∪ 𝑥 ∈ ({1} ∩ 𝑍)ran (𝑅𝑟𝑥))) ∪ ( 𝑥𝑍 dom (𝑅𝑟𝑥) ∪ 𝑥𝑍 ran (𝑅𝑟𝑥))) = (dom 𝑅 ∪ ran 𝑅))
18235, 181syl5eq 2806 . . 3 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅))
183 nn0ex 11490 . . . . . . 7 0 ∈ V
184183ssex 4954 . . . . . 6 (𝑍 ⊆ ℕ0𝑍 ∈ V)
185 incom 3948 . . . . . . . . . 10 (𝑍 ∩ {0}) = ({0} ∩ 𝑍)
186 inex1g 4953 . . . . . . . . . 10 (𝑍 ∈ V → (𝑍 ∩ {0}) ∈ V)
187185, 186syl5eqelr 2844 . . . . . . . . 9 (𝑍 ∈ V → ({0} ∩ 𝑍) ∈ V)
188 incom 3948 . . . . . . . . . 10 (𝑍 ∩ {1}) = ({1} ∩ 𝑍)
189 inex1g 4953 . . . . . . . . . 10 (𝑍 ∈ V → (𝑍 ∩ {1}) ∈ V)
190188, 189syl5eqelr 2844 . . . . . . . . 9 (𝑍 ∈ V → ({1} ∩ 𝑍) ∈ V)
191 unexg 7124 . . . . . . . . 9 ((({0} ∩ 𝑍) ∈ V ∧ ({1} ∩ 𝑍) ∈ V) → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
192187, 190, 191syl2anc 696 . . . . . . . 8 (𝑍 ∈ V → (({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V)
193 unexg 7124 . . . . . . . 8 (((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∈ V ∧ 𝑍 ∈ V) → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
194192, 193mpancom 706 . . . . . . 7 (𝑍 ∈ V → ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V)
195 ovex 6841 . . . . . . . 8 (𝑅𝑟𝑥) ∈ V
196195rgenw 3062 . . . . . . 7 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V
197 iunexg 7308 . . . . . . 7 ((((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍) ∈ V ∧ ∀𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V) → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
198194, 196, 197sylancl 697 . . . . . 6 (𝑍 ∈ V → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
199184, 198syl 17 . . . . 5 (𝑍 ⊆ ℕ0 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
2001993ad2ant2 1129 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V)
201 simp1 1131 . . . 4 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → 𝑅𝑉)
202 relexp0eq 38495 . . . 4 (( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∈ V ∧ 𝑅𝑉) → ((dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅) ↔ ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0)))
203200, 201, 202syl2anc 696 . . 3 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ((dom 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥) ∪ ran 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)) = (dom 𝑅 ∪ ran 𝑅) ↔ ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0)))
204182, 203mpbid 222 . 2 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥 ∈ ((({0} ∩ 𝑍) ∪ ({1} ∩ 𝑍)) ∪ 𝑍)(𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
20512, 204syl5eq 2806 1 ((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  Vcvv 3340  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321  {cpr 4323   ciun 4672   I cid 5173  dom cdm 5266  ran crn 5267  cres 5268  (class class class)co 6813  0cc0 10128  1c1 10129  0cn0 11484  𝑟crelexp 13959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-seq 12996  df-relexp 13960
This theorem is referenced by:  corclrcl  38501  corcltrcl  38533
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