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Theorem dfrtrcl5 38253
Description: Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
Assertion
Ref Expression
dfrtrcl5 t* = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))})
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfrtrcl5
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rtrcl 13773 . 2 t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
2 ancom 465 . . . . . . 7 ((( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦) ↔ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
32anbi2i 730 . . . . . 6 ((𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)) ↔ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)))
43abbii 2768 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))} = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
54inteqi 4511 . . . 4 {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))} = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
65mpteq2i 4774 . . 3 (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))}) = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
7 vex 3234 . . . . . 6 𝑥 ∈ V
87rtrclexi 38245 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V
98a1i 11 . . . 4 (𝑥 ∈ V → {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)
10 dmexg 7139 . . . . . . . . 9 (𝑥 ∈ V → dom 𝑥 ∈ V)
11 rnexg 7140 . . . . . . . . 9 (𝑥 ∈ V → ran 𝑥 ∈ V)
12 unexg 7001 . . . . . . . . 9 ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → (dom 𝑥 ∪ ran 𝑥) ∈ V)
1310, 11, 12syl2anc 694 . . . . . . . 8 (𝑥 ∈ V → (dom 𝑥 ∪ ran 𝑥) ∈ V)
14 resiexg 7144 . . . . . . . 8 ((dom 𝑥 ∪ ran 𝑥) ∈ V → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
157, 13, 14mp2b 10 . . . . . . 7 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
167, 15unex 6998 . . . . . 6 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
1716trclexi 38244 . . . . 5 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V
1817a1i 11 . . . 4 (𝑥 ∈ V → {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
19 simpr 476 . . . . . 6 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) → (𝑧𝑧) ⊆ 𝑧)
2019cotrintab 38238 . . . . 5 ( {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∘ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
2120a1i 11 . . . 4 (𝑥 ∈ V → ( {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∘ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
227dmex 7141 . . . . . . . . . . . . 13 dom 𝑥 ∈ V
237rnex 7142 . . . . . . . . . . . . 13 ran 𝑥 ∈ V
2412resiexd 6521 . . . . . . . . . . . . 13 ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
2522, 23, 24mp2an 708 . . . . . . . . . . . 12 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
267, 25unex 6998 . . . . . . . . . . 11 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
27 dmtrcl 38251 . . . . . . . . . . 11 ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V → dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))))
2826, 27ax-mp 5 . . . . . . . . . 10 dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
29 dmun 5363 . . . . . . . . . . 11 dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
30 dmresi 5492 . . . . . . . . . . . 12 dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3130uneq2i 3797 . . . . . . . . . . 11 (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
32 ssun1 3809 . . . . . . . . . . . 12 dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
33 ssequn1 3816 . . . . . . . . . . . 12 (dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
3432, 33mpbi 220 . . . . . . . . . . 11 (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3529, 31, 343eqtri 2677 . . . . . . . . . 10 dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
3628, 35eqtri 2673 . . . . . . . . 9 dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
37 rntrcl 38252 . . . . . . . . . . 11 ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V → ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))))
3826, 37ax-mp 5 . . . . . . . . . 10 ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
39 rnun 5576 . . . . . . . . . . 11 ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
40 rnresi 5514 . . . . . . . . . . . 12 ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4140uneq2i 3797 . . . . . . . . . . 11 (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
42 ssun2 3810 . . . . . . . . . . . 12 ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
43 ssequn1 3816 . . . . . . . . . . . 12 (ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
4442, 43mpbi 220 . . . . . . . . . . 11 (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4539, 41, 443eqtri 2677 . . . . . . . . . 10 ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
4638, 45eqtri 2673 . . . . . . . . 9 ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
4736, 46uneq12i 3798 . . . . . . . 8 (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥))
48 unidm 3789 . . . . . . . 8 ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4947, 48eqtri 2673 . . . . . . 7 (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (dom 𝑥 ∪ ran 𝑥)
5049reseq2i 5425 . . . . . 6 ( I ↾ (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
51 ssun2 3810 . . . . . . 7 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
52 ssmin 4528 . . . . . . 7 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
5351, 52sstri 3645 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
5450, 53eqsstri 3668 . . . . 5 ( I ↾ (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
5554a1i 11 . . . 4 (𝑥 ∈ V → ( I ↾ (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
56 simprl 809 . . . . . 6 ((𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑦𝑦) ⊆ 𝑦)
5756cotrintab 38238 . . . . 5 ( {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) ⊆ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
5857a1i 11 . . . 4 (𝑥 ∈ V → ( {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) ⊆ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
59 id 22 . . . . . 6 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → 𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
6059, 59coeq12d 5319 . . . . 5 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (𝑦𝑦) = ( {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∘ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}))
6160, 59sseq12d 3667 . . . 4 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → ((𝑦𝑦) ⊆ 𝑦 ↔ ( {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∘ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}))
62 dmeq 5356 . . . . . . 7 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → dom 𝑦 = dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
63 rneq 5383 . . . . . . 7 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → ran 𝑦 = ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
6462, 63uneq12d 3801 . . . . . 6 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (dom 𝑦 ∪ ran 𝑦) = (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}))
6564reseq2d 5428 . . . . 5 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})))
6665, 59sseq12d 3667 . . . 4 (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∪ ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}))
67 id 22 . . . . . 6 (𝑧 = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → 𝑧 = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
6867, 67coeq12d 5319 . . . . 5 (𝑧 = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝑧𝑧) = ( {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}))
6968, 67sseq12d 3667 . . . 4 (𝑧 = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → ((𝑧𝑧) ⊆ 𝑧 ↔ ( {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) ⊆ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}))
709, 18, 21, 55, 58, 61, 66, 69mptrcllem 38237 . . 3 (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ V ↦ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
71 df-3an 1056 . . . . . . 7 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ∧ (𝑧𝑧) ⊆ 𝑧))
72 ancom 465 . . . . . . . . 9 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ↔ (𝑥𝑧 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧))
73 unss 3820 . . . . . . . . 9 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
7472, 73bitri 264 . . . . . . . 8 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
7574anbi1i 731 . . . . . . 7 (((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ∧ (𝑧𝑧) ⊆ 𝑧) ↔ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧))
7671, 75bitr2i 265 . . . . . 6 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧))
7776abbii 2768 . . . . 5 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
7877inteqi 4511 . . . 4 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
7978mpteq2i 4774 . . 3 (𝑥 ∈ V ↦ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
806, 70, 793eqtri 2677 . 2 (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))}) = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
811, 80eqtr4i 2676 1 t* = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  cun 3605  wss 3607   cint 4507  cmpt 4762   I cid 5052  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147  t*crtcl 13771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-rtrcl 13773
This theorem is referenced by: (None)
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