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Theorem dfrtrcl5 42149
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 14917 . 2 t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
2 ancom 461 . . . . . . 7 ((( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦) ↔ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
32anbi2i 623 . . . . . 6 ((𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)) ↔ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)))
43abbii 2801 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))} = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
54inteqi 4947 . . . 4 {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))} = {𝑦 ∣ (𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
65mpteq2i 5246 . . 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 3477 . . . . . 6 𝑥 ∈ V
87rtrclexi 42141 . . . . 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 7876 . . . . . . . . 9 (𝑥 ∈ V → dom 𝑥 ∈ V)
11 rnexg 7877 . . . . . . . . 9 (𝑥 ∈ V → ran 𝑥 ∈ V)
1210, 11unexd 7724 . . . . . . . 8 (𝑥 ∈ V → (dom 𝑥 ∪ ran 𝑥) ∈ V)
13 resiexg 7887 . . . . . . . 8 ((dom 𝑥 ∪ ran 𝑥) ∈ V → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
147, 12, 13mp2b 10 . . . . . . 7 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
157, 14unex 7716 . . . . . 6 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
1615trclexi 42140 . . . . 5 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V
1716a1i 11 . . . 4 (𝑥 ∈ V → {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
18 simpr 485 . . . . . 6 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) → (𝑧𝑧) ⊆ 𝑧)
1918cotrintab 42134 . . . . 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)}
2019a1i 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)})
217dmex 7884 . . . . . . . . . . . . 13 dom 𝑥 ∈ V
227rnex 7885 . . . . . . . . . . . . 13 ran 𝑥 ∈ V
23 unexg 7719 . . . . . . . . . . . . . 14 ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → (dom 𝑥 ∪ ran 𝑥) ∈ V)
2423resiexd 7202 . . . . . . . . . . . . 13 ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
2521, 22, 24mp2an 690 . . . . . . . . . . . 12 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
267, 25unex 7716 . . . . . . . . . . 11 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
27 dmtrcl 42147 . . . . . . . . . . 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 5902 . . . . . . . . . . 11 dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
30 dmresi 6041 . . . . . . . . . . . 12 dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3130uneq2i 4156 . . . . . . . . . . 11 (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
32 ssun1 4168 . . . . . . . . . . . 12 dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
33 ssequn1 4176 . . . . . . . . . . . 12 (dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
3432, 33mpbi 229 . . . . . . . . . . 11 (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3529, 31, 343eqtri 2763 . . . . . . . . . 10 dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
3628, 35eqtri 2759 . . . . . . . . 9 dom {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
37 rntrcl 42148 . . . . . . . . . . 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 6134 . . . . . . . . . . 11 ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
40 rnresi 6063 . . . . . . . . . . . 12 ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4140uneq2i 4156 . . . . . . . . . . 11 (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
42 ssun2 4169 . . . . . . . . . . . 12 ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
43 ssequn1 4176 . . . . . . . . . . . 12 (ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
4442, 43mpbi 229 . . . . . . . . . . 11 (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4539, 41, 443eqtri 2763 . . . . . . . . . 10 ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
4638, 45eqtri 2759 . . . . . . . . 9 ran {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
4736, 46uneq12i 4157 . . . . . . . 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 4148 . . . . . . . 8 ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
4947, 48eqtri 2759 . . . . . . 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 5970 . . . . . 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 4169 . . . . . . 7 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
52 ssmin 4964 . . . . . . 7 (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
5351, 52sstri 3987 . . . . . 6 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
5450, 53eqsstri 4012 . . . . 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 769 . . . . . 6 ((𝑥𝑦 ∧ ((𝑦𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑦𝑦) ⊆ 𝑦)
5756cotrintab 42134 . . . . 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 5856 . . . . 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 4011 . . . 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 5895 . . . . . . 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 5927 . . . . . . 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 4160 . . . . . 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 5973 . . . . 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 4011 . . . 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 5856 . . . . 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 4011 . . . 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, 17, 20, 55, 58, 61, 66, 69mptrcllem 42133 . . 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 1089 . . . . . . 7 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ∧ (𝑧𝑧) ⊆ 𝑧))
72 ancom 461 . . . . . . . . 9 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ↔ (𝑥𝑧 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧))
73 unss 4180 . . . . . . . . 9 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
7472, 73bitri 274 . . . . . . . 8 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
7574anbi1i 624 . . . . . . 7 (((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧) ∧ (𝑧𝑧) ⊆ 𝑧) ↔ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧))
7671, 75bitr2i 275 . . . . . 6 (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧))
7776abbii 2801 . . . . 5 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
7877inteqi 4947 . . . 4 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
7978mpteq2i 5246 . . 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 2763 . 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 2762 1 t* = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2708  Vcvv 3473  cun 3942  wss 3944   cint 4943  cmpt 5224   I cid 5566  dom cdm 5669  ran crn 5670  cres 5671  ccom 5673  t*crtcl 14915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-rtrcl 14917
This theorem is referenced by: (None)
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