Theorem rtrclexi 38454

Description: The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)

Hypothesis

Ref	Expression
rtrclexi.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
rtrclexi	⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rtrclexi

Step	Hyp	Ref	Expression
1		ssun1 3927	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
2		coundir 5781	. . . . 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 5780	. . . . . . 7 ⊢ (𝐴 ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) = ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
4		cossxp 5802	. . . . . . . . 9 ⊢ (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5		ssun1 3927	. . . . . . . . . 10 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		ssun2 3928	. . . . . . . . . 10 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
7		xpss12 5264	. . . . . . . . . 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)))
8	5, 6, 7	mp2an 672	. . . . . . . . 9 ⊢ (dom 𝐴 × ran 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
9	4, 8	sstri 3761	. . . . . . . 8 ⊢ (𝐴 ∘ 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
10		cossxp 5802	. . . . . . . . 9 ⊢ (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) × ran 𝐴)
11		dmxpss 5706	. . . . . . . . . 10 ⊢ dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom 𝐴 ∪ ran 𝐴)
12		xpss12 5264	. . . . . . . . . 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)))
13	11, 6, 12	mp2an 672	. . . . . . . . 9 ⊢ (dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) × ran 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
14	10, 13	sstri 3761	. . . . . . . 8 ⊢ (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
15	9, 14	unssi 3939	. . . . . . 7 ⊢ ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
16	3, 15	eqsstri 3784	. . . . . 6 ⊢ (𝐴 ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
17		coundi 5780	. . . . . . 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 5802	. . . . . . . . 9 ⊢ (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ 𝐴) ⊆ (dom 𝐴 × ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
19		rnxpss 5707	. . . . . . . . . 10 ⊢ ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom 𝐴 ∪ ran 𝐴)
20		xpss12 5264	. . . . . . . . . 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)))
21	5, 19, 20	mp2an 672	. . . . . . . . 9 ⊢ (dom 𝐴 × ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
22	18, 21	sstri 3761	. . . . . . . 8 ⊢ (((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∘ 𝐴) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
23		xpidtr 5659	. . . . . . . 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))
24	22, 23	unssi 3939	. . . . . . 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))
25	17, 24	eqsstri 3784	. . . . . 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))
26	16, 25	unssi 3939	. . . . 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))
27	2, 26	eqsstri 3784	. . . 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 3928	. . . 4 ⊢ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ⊆ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
29	27, 28	sstri 3761	. . 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 5469	. . . . . . . 8 ⊢ dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
31	5, 11	unssi 3939	. . . . . . . 8 ⊢ (dom 𝐴 ∪ dom ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
32	30, 31	eqsstri 3784	. . . . . . 7 ⊢ dom (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
33		rnun 5682	. . . . . . . 8 ⊢ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))
34	6, 19	unssi 3939	. . . . . . . 8 ⊢ (ran 𝐴 ∪ ran ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
35	33, 34	eqsstri 3784	. . . . . . 7 ⊢ ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ⊆ (dom 𝐴 ∪ ran 𝐴)
36	32, 35	unssi 3939	. . . . . 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 5566	. . . . . 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)))
38	36, 37	ax-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 5567	. . . . . . 7 ⊢ Rel ( I ↾ (dom 𝐴 ∪ ran 𝐴))
40		relssdmrn 5800	. . . . . . 7 ⊢ (Rel ( I ↾ (dom 𝐴 ∪ ran 𝐴)) → ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) × ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
41	39, 40	ax-mp 5	. . . . . 6 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) × ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
42		dmresi 5598	. . . . . . 7 ⊢ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
43		rnresi 5620	. . . . . . 7 ⊢ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
44	42, 43	xpeq12i 5277	. . . . . 6 ⊢ (dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) × ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
45	41, 44	sseqtri 3786	. . . . 5 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))
46	38, 45	sstri 3761	. . . 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))
47	46, 28	sstri 3761	. . 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)))))
49	29, 47, 48	mp2an 672	. 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 ⊢ 𝐴 ∈ 𝑉
51	50	elexi 3365	. . . . 5 ⊢ 𝐴 ∈ V
52	51	dmex 7246	. . . . . . 7 ⊢ dom 𝐴 ∈ V
53	51	rnex 7247	. . . . . . 7 ⊢ ran 𝐴 ∈ V
54	52, 53	unex 7103	. . . . . 6 ⊢ (dom 𝐴 ∪ ran 𝐴) ∈ V
55	54, 54	xpex 7109	. . . . 5 ⊢ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)) ∈ V
56	51, 55	unex 7103	. . . 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))))
58	57, 57	coeq12d 5425	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → (𝑥 ∘ 𝑥) = ((𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) ∘ (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴)))))
59	58, 57	sseq12d 3783	. . . . . 6 ⊢ (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → ((𝑥 ∘ 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)))))
60		dmeq 5462	. . . . . . . . 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 5489	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → ran 𝑥 = ran (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))))
62	60, 61	uneq12d 3919	. . . . . . . 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)))))
63	62	reseq2d 5534	. . . . . . 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))))))
64	63, 57	sseq12d 3783	. . . . . 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)))))
65	59, 64	anbi12d 616	. . . . 5 ⊢ (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → (((𝑥 ∘ 𝑥) ⊆ 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))))))
66	65	cleq2lem 38440	. . . 4 ⊢ (𝑥 = (𝐴 ∪ ((dom 𝐴 ∪ ran 𝐴) × (dom 𝐴 ∪ ran 𝐴))) → ((𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 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)))))))
67	56, 66	spcev 3451	. . 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(𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)))
68		intexab 4953	. . 3 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)
69	67, 68	sylib 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 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)
70	1, 49, 69	mp2an 672	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V

Syntax hints:   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  Vcvv 3351   ∪ cun 3721   ⊆ wss 3723  ∩ cint 4611   I cid 5156   × cxp 5247  dom cdm 5249  ran crn 5250   ↾ cres 5251   ∘ ccom 5253  Rel wrel 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262

This theorem is referenced by:  dfrtrcl5  38462