Theorem dfrtrcl5 42938

Description: Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)

Assertion

Ref	Expression
dfrtrcl5	⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))})

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfrtrcl5

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rtrcl 14938	. 2 ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
2		ancom 460	. . . . . . 7 ⊢ ((( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦) ↔ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))
3	2	anbi2i 622	. . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)) ↔ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)))
4	3	abbii 2796	. . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))} = {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
5	4	inteqi 4947	. . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))} = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
6	5	mpteq2i 5246	. . 3 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))}) = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
7		vex 3472	. . . . . 6 ⊢ 𝑥 ∈ V
8	7	rtrclexi 42930	. . . . 5 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V
9	8	a1i 11	. . . 4 ⊢ (𝑥 ∈ V → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)
10		dmexg 7890	. . . . . . . . 9 ⊢ (𝑥 ∈ V → dom 𝑥 ∈ V)
11		rnexg 7891	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V)
12	10, 11	unexd 7737	. . . . . . . 8 ⊢ (𝑥 ∈ V → (dom 𝑥 ∪ ran 𝑥) ∈ V)
13		resiexg 7901	. . . . . . . 8 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∈ V → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
14	7, 12, 13	mp2b 10	. . . . . . 7 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
15	7, 14	unex 7729	. . . . . 6 ⊢ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
16	15	trclexi 42929	. . . . 5 ⊢ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V
17	16	a1i 11	. . . 4 ⊢ (𝑥 ∈ V → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V)
18		simpr 484	. . . . . 6 ⊢ (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) → (𝑧 ∘ 𝑧) ⊆ 𝑧)
19	18	cotrintab 42923	. . . . 5 ⊢ (∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∘ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
20	19	a1i 11	. . . 4 ⊢ (𝑥 ∈ V → (∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∘ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
21	7	dmex 7898	. . . . . . . . . . . . 13 ⊢ dom 𝑥 ∈ V
22	7	rnex 7899	. . . . . . . . . . . . 13 ⊢ ran 𝑥 ∈ V
23		unexg 7732	. . . . . . . . . . . . . 14 ⊢ ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → (dom 𝑥 ∪ ran 𝑥) ∈ V)
24	23	resiexd 7212	. . . . . . . . . . . . 13 ⊢ ((dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
25	21, 22, 24	mp2an 689	. . . . . . . . . . . 12 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
26	7, 25	unex 7729	. . . . . . . . . . 11 ⊢ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V
27		dmtrcl 42936	. . . . . . . . . . 11 ⊢ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V → dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))))
28	26, 27	ax-mp 5	. . . . . . . . . 10 ⊢ dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
29		dmun 5903	. . . . . . . . . . 11 ⊢ dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
30		dmresi 6044	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
31	30	uneq2i 4155	. . . . . . . . . . 11 ⊢ (dom 𝑥 ∪ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
32		ssun1 4167	. . . . . . . . . . . 12 ⊢ dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
33		ssequn1 4175	. . . . . . . . . . . 12 ⊢ (dom 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
34	32, 33	mpbi 229	. . . . . . . . . . 11 ⊢ (dom 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
35	29, 31, 34	3eqtri 2758	. . . . . . . . . 10 ⊢ dom (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
36	28, 35	eqtri 2754	. . . . . . . . 9 ⊢ dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
37		rntrcl 42937	. . . . . . . . . . 11 ⊢ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ∈ V → ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))))
38	26, 37	ax-mp 5	. . . . . . . . . 10 ⊢ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
39		rnun 6138	. . . . . . . . . . 11 ⊢ ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)))
40		rnresi 6067	. . . . . . . . . . . 12 ⊢ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
41	40	uneq2i 4155	. . . . . . . . . . 11 ⊢ (ran 𝑥 ∪ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥))
42		ssun2 4168	. . . . . . . . . . . 12 ⊢ ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥)
43		ssequn1 4175	. . . . . . . . . . . 12 ⊢ (ran 𝑥 ⊆ (dom 𝑥 ∪ ran 𝑥) ↔ (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥))
44	42, 43	mpbi 229	. . . . . . . . . . 11 ⊢ (ran 𝑥 ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
45	39, 41, 44	3eqtri 2758	. . . . . . . . . 10 ⊢ ran (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) = (dom 𝑥 ∪ ran 𝑥)
46	38, 45	eqtri 2754	. . . . . . . . 9 ⊢ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = (dom 𝑥 ∪ ran 𝑥)
47	36, 46	uneq12i 4156	. . . . . . . 8 ⊢ (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) = ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥))
48		unidm 4147	. . . . . . . 8 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
49	47, 48	eqtri 2754	. . . . . . 7 ⊢ (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) = (dom 𝑥 ∪ ran 𝑥)
50	49	reseq2i 5971	. . . . . 6 ⊢ ( I ↾ (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
51		ssun2 4168	. . . . . . 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 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
53	51, 52	sstri 3986	. . . . . 6 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
54	50, 53	eqsstri 4011	. . . . 5 ⊢ ( I ↾ (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})) ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
55	54	a1i 11	. . . 4 ⊢ (𝑥 ∈ V → ( I ↾ (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})) ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
56		simprl 768	. . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)) → (𝑦 ∘ 𝑦) ⊆ 𝑦)
57	56	cotrintab 42923	. . . . 5 ⊢ (∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) ⊆ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}
58	57	a1i 11	. . . 4 ⊢ (𝑥 ∈ V → (∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) ⊆ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
59		id 22	. . . . . 6 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → 𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
60	59, 59	coeq12d 5857	. . . . 5 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (𝑦 ∘ 𝑦) = (∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∘ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}))
61	60, 59	sseq12d 4010	. . . 4 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → ((𝑦 ∘ 𝑦) ⊆ 𝑦 ↔ (∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∘ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}))
62		dmeq 5896	. . . . . . 7 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → dom 𝑦 = dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
63		rneq 5928	. . . . . . 7 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → ran 𝑦 = ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
64	62, 63	uneq12d 4159	. . . . . 6 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (dom 𝑦 ∪ ran 𝑦) = (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}))
65	64	reseq2d 5974	. . . . 5 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → ( I ↾ (dom 𝑦 ∪ ran 𝑦)) = ( I ↾ (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})))
66	65, 59	sseq12d 4010	. . . 4 ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ ( I ↾ (dom ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∪ ran ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})) ⊆ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}))
67		id 22	. . . . . 6 ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → 𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))})
68	67, 67	coeq12d 5857	. . . . 5 ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝑧 ∘ 𝑧) = (∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}))
69	68, 67	sseq12d 4010	. . . 4 ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∘ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) ⊆ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}))
70	9, 17, 20, 55, 58, 61, 66, 69	mptrcllem 42922	. . 3 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ((𝑦 ∘ 𝑦) ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
71		df-3an 1086	. . . . . . 7 ⊢ ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧) ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧))
72		ancom 460	. . . . . . . . 9 ⊢ ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧) ↔ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧))
73		unss 4179	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
74	72, 73	bitri 275	. . . . . . . 8 ⊢ ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧) ↔ (𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧)
75	74	anbi1i 623	. . . . . . 7 ⊢ (((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧) ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧))
76	71, 75	bitr2i 276	. . . . . 6 ⊢ (((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧))
77	76	abbii 2796	. . . . 5 ⊢ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
78	77	inteqi 4947	. . . 4 ⊢ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
79	78	mpteq2i 5246	. . 3 ⊢ (𝑥 ∈ V ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
80	6, 70, 79	3eqtri 2758	. 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))}) = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
81	1, 80	eqtr4i 2757	1 ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))})

Syntax hints:   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  Vcvv 3468   ∪ cun 3941   ⊆ wss 3943  ∩ cint 4943   ↦ cmpt 5224   I cid 5566  dom cdm 5669  ran crn 5670   ↾ cres 5671   ∘ ccom 5673  t*crtcl 14936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-rtrcl 14938

This theorem is referenced by: (None)