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Theorem dfrtrcl5 40703
 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 14396 . 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 626 . . . . . 6 ((𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)) ↔ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)))
43abbii 2824 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))} = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
54inteqi 4843 . . . 4 {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))} = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
65mpteq2i 5125 . . 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 3414 . . . . . 6 𝑥 ∈ V
87rtrclexi 40695 . . . . 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 7614 . . . . . . . . 9 (𝑥 ∈ V → dom 𝑥 ∈ V)
11 rnexg 7615 . . . . . . . . 9 (𝑥 ∈ V → ran 𝑥 ∈ V)
12 unexg 7471 . . . . . . . . 9 ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → (dom 𝑥 ∪ ran 𝑥) ∈ V)
1310, 11, 12syl2anc 588 . . . . . . . 8 (𝑥 ∈ V → (dom 𝑥 ∪ ran 𝑥) ∈ V)
14 resiexg 7625 . . . . . . . 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 7468 . . . . . 6 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
1716trclexi 40694 . . . . 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 489 . . . . . 6 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) → (𝑧𝑧) ⊆ 𝑧)
2019cotrintab 40688 . . . . 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 7622 . . . . . . . . . . . . 13 dom 𝑥 ∈ V
237rnex 7623 . . . . . . . . . . . . 13 ran 𝑥 ∈ V
2412resiexd 6971 . . . . . . . . . . . . 13 ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
2522, 23, 24mp2an 692 . . . . . . . . . . . 12 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
267, 25unex 7468 . . . . . . . . . . 11 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
27 dmtrcl 40701 . . . . . . . . . . 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 5751 . . . . . . . . . . 11 dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
30 dmresi 5894 . . . . . . . . . . . 12 dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3130uneq2i 4066 . . . . . . . . . . 11 (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
32 ssun1 4078 . . . . . . . . . . . 12 dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
33 ssequn1 4086 . . . . . . . . . . . 12 (dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
3432, 33mpbi 233 . . . . . . . . . . 11 (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3529, 31, 343eqtri 2786 . . . . . . . . . 10 dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
3628, 35eqtri 2782 . . . . . . . . 9 dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
37 rntrcl 40702 . . . . . . . . . . 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 5977 . . . . . . . . . . 11 ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
40 rnresi 5916 . . . . . . . . . . . 12 ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4140uneq2i 4066 . . . . . . . . . . 11 (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
42 ssun2 4079 . . . . . . . . . . . 12 ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
43 ssequn1 4086 . . . . . . . . . . . 12 (ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
4442, 43mpbi 233 . . . . . . . . . . 11 (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4539, 41, 443eqtri 2786 . . . . . . . . . 10 ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
4638, 45eqtri 2782 . . . . . . . . 9 ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
4736, 46uneq12i 4067 . . . . . . . 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 4058 . . . . . . . 8 ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4947, 48eqtri 2782 . . . . . . 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 5821 . . . . . 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 4079 . . . . . . 7 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
52 ssmin 4858 . . . . . . 7 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
5351, 52sstri 3902 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
5450, 53eqsstri 3927 . . . . 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 771 . . . . . 6 ((𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑦𝑦) ⊆ 𝑦)
5756cotrintab 40688 . . . . 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 5705 . . . . 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 3926 . . . 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 5744 . . . . . . 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 5778 . . . . . . 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 4070 . . . . . 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 5824 . . . . 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 3926 . . . 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 5705 . . . . 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 3926 . . . 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 40687 . . 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 1087 . . . . . . 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 4090 . . . . . . . . 9 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
7472, 73bitri 278 . . . . . . . 8 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
7574anbi1i 627 . . . . . . 7 (((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ∧ (𝑧𝑧) ⊆ 𝑧) ↔ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧))
7671, 75bitr2i 279 . . . . . 6 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧))
7776abbii 2824 . . . . 5 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
7877inteqi 4843 . . . 4 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
7978mpteq2i 5125 . . 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 2786 . 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 2785 1 t* = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))})
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  {cab 2736  Vcvv 3410   ∪ cun 3857   ⊆ wss 3859  ∩ cint 4839   ↦ cmpt 5113   I cid 5430  dom cdm 5525  ran crn 5526   ↾ cres 5527   ∘ ccom 5529  t*crtcl 14394 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-rtrcl 14396 This theorem is referenced by: (None)
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