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Theorem rtrclexi 38430
Description: The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
Hypothesis
Ref Expression
rtrclexi.1 𝐴𝑉
Assertion
Ref Expression
rtrclexi {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rtrclexi
StepHypRef Expression
1 ssun1 3919 . 2 𝐴 ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
2 coundir 5798 . . . . 5 ((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) = ((𝐴 ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ∪ (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))
3 coundi 5797 . . . . . . 7 (𝐴 ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) = ((𝐴𝐴) ∪ (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
4 cossxp 5819 . . . . . . . . 9 (𝐴𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5 ssun1 3919 . . . . . . . . . 10 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 ssun2 3920 . . . . . . . . . 10 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
7 xpss12 5281 . . . . . . . . . 10 ((dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ∧ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)) → (dom 𝐴 × ran 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
85, 6, 7mp2an 710 . . . . . . . . 9 (dom 𝐴 × ran 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
94, 8sstri 3753 . . . . . . . 8 (𝐴𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
10 cossxp 5819 . . . . . . . . 9 (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) × ran 𝐴)
11 dmxpss 5723 . . . . . . . . . 10 dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom 𝐴 ∪ ran 𝐴)
12 xpss12 5281 . . . . . . . . . 10 ((dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom 𝐴 ∪ ran 𝐴) ∧ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)) → (dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) × ran 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
1311, 6, 12mp2an 710 . . . . . . . . 9 (dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) × ran 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
1410, 13sstri 3753 . . . . . . . 8 (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
159, 14unssi 3931 . . . . . . 7 ((𝐴𝐴) ∪ (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
163, 15eqsstri 3776 . . . . . 6 (𝐴 ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
17 coundi 5797 . . . . . . 7 (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) = ((((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ 𝐴) ∪ (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
18 cossxp 5819 . . . . . . . . 9 (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ 𝐴) ⊆ (dom 𝐴 × ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
19 rnxpss 5724 . . . . . . . . . 10 ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom 𝐴 ∪ ran 𝐴)
20 xpss12 5281 . . . . . . . . . 10 ((dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ∧ ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom 𝐴 ∪ ran 𝐴)) → (dom 𝐴 × ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
215, 19, 20mp2an 710 . . . . . . . . 9 (dom 𝐴 × ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
2218, 21sstri 3753 . . . . . . . 8 (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
23 xpidtr 5676 . . . . . . . 8 (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
2422, 23unssi 3931 . . . . . . 7 ((((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ 𝐴) ∪ (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
2517, 24eqsstri 3776 . . . . . 6 (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
2616, 25unssi 3931 . . . . 5 ((𝐴 ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ∪ (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
272, 26eqsstri 3776 . . . 4 ((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
28 ssun2 3920 . . . 4 ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
2927, 28sstri 3753 . . 3 ((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
30 dmun 5486 . . . . . . . 8 dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
315, 11unssi 3931 . . . . . . . 8 (dom 𝐴 ∪ dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
3230, 31eqsstri 3776 . . . . . . 7 dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
33 rnun 5699 . . . . . . . 8 ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
346, 19unssi 3931 . . . . . . . 8 (ran 𝐴 ∪ ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
3533, 34eqsstri 3776 . . . . . . 7 ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
3632, 35unssi 3931 . . . . . 6 (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (dom 𝐴 ∪ ran 𝐴)
37 ssres2 5583 . . . . . 6 ((dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (dom 𝐴 ∪ ran 𝐴) → ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
3836, 37ax-mp 5 . . . . 5 ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴))
39 relres 5584 . . . . . . 7 Rel ( I ↾ (dom 𝐴 ∪ ran 𝐴))
40 relssdmrn 5817 . . . . . . 7 (Rel ( I ↾ (dom 𝐴 ∪ ran 𝐴)) → ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) × ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
4139, 40ax-mp 5 . . . . . 6 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) × ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
42 dmresi 5615 . . . . . . 7 dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
43 rnresi 5637 . . . . . . 7 ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
4442, 43xpeq12i 5294 . . . . . 6 (dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) × ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
4541, 44sseqtri 3778 . . . . 5 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
4638, 45sstri 3753 . . . 4 ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
4746, 28sstri 3753 . . 3 ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
48 id 22 . . 3 ((((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) → (((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))
4929, 47, 48mp2an 710 . 2 (((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
50 rtrclexi.1 . . . . . 6 𝐴𝑉
5150elexi 3353 . . . . 5 𝐴 ∈ V
5251dmex 7264 . . . . . . 7 dom 𝐴 ∈ V
5351rnex 7265 . . . . . . 7 ran 𝐴 ∈ V
5452, 53unex 7121 . . . . . 6 (dom 𝐴 ∪ ran 𝐴) ∈ V
5554, 54xpex 7127 . . . . 5 ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∈ V
5651, 55unex 7121 . . . 4 (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∈ V
57 id 22 . . . . . . . 8 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → 𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
5857, 57coeq12d 5442 . . . . . . 7 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → (𝑥𝑥) = ((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))
5958, 57sseq12d 3775 . . . . . 6 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → ((𝑥𝑥) ⊆ 𝑥 ↔ ((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))
60 dmeq 5479 . . . . . . . . 9 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → dom 𝑥 = dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
61 rneq 5506 . . . . . . . . 9 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → ran 𝑥 = ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
6260, 61uneq12d 3911 . . . . . . . 8 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))
6362reseq2d 5551 . . . . . . 7 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))))
6463, 57sseq12d 3775 . . . . . 6 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))
6559, 64anbi12d 749 . . . . 5 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → (((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥) ↔ (((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))))
6665cleq2lem 38416 . . . 4 (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → ((𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)) ↔ (𝐴 ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ (((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))))
6756, 66spcev 3440 . . 3 ((𝐴 ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ (((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) → ∃𝑥(𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)))
68 intexab 4971 . . 3 (∃𝑥(𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)) ↔ {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)
6967, 68sylib 208 . 2 ((𝐴 ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ (((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))) → {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)
701, 49, 69mp2an 710 1 {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  Vcvv 3340  cun 3713  wss 3715   cint 4627   I cid 5173   × cxp 5264  dom cdm 5266  ran crn 5267  cres 5268  ccom 5270  Rel wrel 5271
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-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279
This theorem is referenced by:  dfrtrcl5  38438
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