Theorem clcnvlem 36949

Description: When 𝐴, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)

Hypotheses

Ref	Expression
clcnvlem.sub1	⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (𝜒 → 𝜓))
clcnvlem.sub2	⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → (𝜓 → 𝜒))
clcnvlem.sub3	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
clcnvlem.ssub	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
clcnvlem.ubex	⊢ (𝜑 → 𝐴 ∈ V)
clcnvlem.clex	⊢ (𝜑 → 𝜃)

Assertion

Ref	Expression
clcnvlem	⊢ (𝜑 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑋   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)

Proof of Theorem clcnvlem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clcnvlem.ubex	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
2		clcnvlem.ssub	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
3		clcnvlem.clex	. . . . 5 ⊢ (𝜑 → 𝜃)
4	2, 3	jca 553	. . . 4 ⊢ (𝜑 → (𝑋 ⊆ 𝐴 ∧ 𝜃))
5		clcnvlem.sub3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
6	5	cleq2lem 36933	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ 𝐴 ∧ 𝜃)))
7	1, 4, 6	elabd 3321	. . 3 ⊢ (𝜑 → ∃𝑥(𝑋 ⊆ 𝑥 ∧ 𝜓))
8	7	cnvintabd 36928	. 2 ⊢ (𝜑 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} = ∩ {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))})
9		df-rab 2905	. . . . 5 ⊢ {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))} = {𝑧 ∣ (𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)))}
10		exsimpl 1783	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) → ∃𝑥 𝑧 = ◡𝑥)
11		relcnv 5422	. . . . . . . . . . . . 13 ⊢ Rel ◡𝑥
12		releq 5124	. . . . . . . . . . . . 13 ⊢ (𝑧 = ◡𝑥 → (Rel 𝑧 ↔ Rel ◡𝑥))
13	11, 12	mpbiri 247	. . . . . . . . . . . 12 ⊢ (𝑧 = ◡𝑥 → Rel 𝑧)
14	13	exlimiv 1845	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑧 = ◡𝑥 → Rel 𝑧)
15	10, 14	syl 17	. . . . . . . . . 10 ⊢ (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) → Rel 𝑧)
16		df-rel 5045	. . . . . . . . . 10 ⊢ (Rel 𝑧 ↔ 𝑧 ⊆ (V × V))
17	15, 16	sylib 207	. . . . . . . . 9 ⊢ (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) → 𝑧 ⊆ (V × V))
18		selpw 4115	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝒫 (V × V) ↔ 𝑧 ⊆ (V × V))
19	18	bicomi 213	. . . . . . . . 9 ⊢ (𝑧 ⊆ (V × V) ↔ 𝑧 ∈ 𝒫 (V × V))
20	17, 19	sylib 207	. . . . . . . 8 ⊢ (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) → 𝑧 ∈ 𝒫 (V × V))
21	20	pm4.71ri 663	. . . . . . 7 ⊢ (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) ↔ (𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))))
22	21	bicomi 213	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))) ↔ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)))
23	22	abbii 2726	. . . . 5 ⊢ {𝑧 ∣ (𝑧 ∈ 𝒫 (V × V) ∧ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)))} = {𝑧 ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))}
24	9, 23	eqtri 2632	. . . 4 ⊢ {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))} = {𝑧 ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))}
25	24	inteqi 4414	. . 3 ⊢ ∩ {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))} = ∩ {𝑧 ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))}
26	25	a1i 11	. 2 ⊢ (𝜑 → ∩ {𝑧 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))} = ∩ {𝑧 ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))})
27		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
28	27	cnvex 7006	. . . . . 6 ⊢ ◡𝑦 ∈ V
29	28	cnvex 7006	. . . . 5 ⊢ ◡◡𝑦 ∈ V
30	29	a1i 11	. . . 4 ⊢ (𝜑 → ◡◡𝑦 ∈ V)
31	1, 2	ssexd 4733	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ V)
32		difexg 4735	. . . . . . . . . . 11 ⊢ (𝑋 ∈ V → (𝑋 ∖ ◡◡𝑋) ∈ V)
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∖ ◡◡𝑋) ∈ V)
34		unexg 6857	. . . . . . . . . 10 ⊢ ((◡𝑦 ∈ V ∧ (𝑋 ∖ ◡◡𝑋) ∈ V) → (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∈ V)
35	28, 33, 34	sylancr 694	. . . . . . . . 9 ⊢ (𝜑 → (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ∈ V)
36		inundif 3998	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = 𝑋
37		cnvun 5457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = (◡(𝑋 ∩ ◡◡𝑋) ∪ ◡(𝑋 ∖ ◡◡𝑋))
38	37	sseq1i 3592	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦 ↔ (◡(𝑋 ∩ ◡◡𝑋) ∪ ◡(𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦)
39	38	biimpi 205	. . . . . . . . . . . . . . . . . . 19 ⊢ (◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦 → (◡(𝑋 ∩ ◡◡𝑋) ∪ ◡(𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦)
40	39	unssad 3752	. . . . . . . . . . . . . . . . . 18 ⊢ (◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦 → ◡(𝑋 ∩ ◡◡𝑋) ⊆ 𝑦)
41		relcnv 5422	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Rel ◡◡𝑋
42		relin2 5160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Rel ◡◡𝑋 → Rel (𝑋 ∩ ◡◡𝑋))
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ Rel (𝑋 ∩ ◡◡𝑋)
44		dfrel2 5502	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel (𝑋 ∩ ◡◡𝑋) ↔ ◡◡(𝑋 ∩ ◡◡𝑋) = (𝑋 ∩ ◡◡𝑋))
45	43, 44	mpbi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ ◡◡(𝑋 ∩ ◡◡𝑋) = (𝑋 ∩ ◡◡𝑋)
46		cnvss 5216	. . . . . . . . . . . . . . . . . . 19 ⊢ (◡(𝑋 ∩ ◡◡𝑋) ⊆ 𝑦 → ◡◡(𝑋 ∩ ◡◡𝑋) ⊆ ◡𝑦)
47	45, 46	syl5eqssr 3613	. . . . . . . . . . . . . . . . . 18 ⊢ (◡(𝑋 ∩ ◡◡𝑋) ⊆ 𝑦 → (𝑋 ∩ ◡◡𝑋) ⊆ ◡𝑦)
48	40, 47	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦 → (𝑋 ∩ ◡◡𝑋) ⊆ ◡𝑦)
49		ssid 3587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∖ ◡◡𝑋) ⊆ (𝑋 ∖ ◡◡𝑋)
50		unss12 3747	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∩ ◡◡𝑋) ⊆ ◡𝑦 ∧ (𝑋 ∖ ◡◡𝑋) ⊆ (𝑋 ∖ ◡◡𝑋)) → ((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
51	48, 49, 50	sylancl 693	. . . . . . . . . . . . . . . 16 ⊢ (◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦 → ((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
52	51	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = 𝑋 → (◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦 → ((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
53		cnveq 5218	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = 𝑋 → ◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = ◡𝑋)
54	53	sseq1d 3595	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = 𝑋 → (◡((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ 𝑦 ↔ ◡𝑋 ⊆ 𝑦))
55		sseq1 3589	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = 𝑋 → (((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) ↔ 𝑋 ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
56	52, 54, 55	3imtr3d 281	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∩ ◡◡𝑋) ∪ (𝑋 ∖ ◡◡𝑋)) = 𝑋 → (◡𝑋 ⊆ 𝑦 → 𝑋 ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
57	36, 56	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (◡𝑋 ⊆ 𝑦 → 𝑋 ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
58		sseq2 3590	. . . . . . . . . . . . 13 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → (𝑋 ⊆ 𝑥 ↔ 𝑋 ⊆ (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))))
59	57, 58	syl5ibr 235	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → (◡𝑋 ⊆ 𝑦 → 𝑋 ⊆ 𝑥))
60	59	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (◡𝑋 ⊆ 𝑦 → 𝑋 ⊆ 𝑥))
61		clcnvlem.sub1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (𝜒 → 𝜓))
62	60, 61	anim12d 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → ((◡𝑋 ⊆ 𝑦 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
63		cnveq 5218	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → ◡𝑥 = ◡(◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)))
64		cnvun 5457	. . . . . . . . . . . . 13 ⊢ ◡(◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) = (◡◡𝑦 ∪ ◡(𝑋 ∖ ◡◡𝑋))
65		cnvnonrel 36913	. . . . . . . . . . . . . . 15 ⊢ ◡(𝑋 ∖ ◡◡𝑋) = ∅
66		0ss 3924	. . . . . . . . . . . . . . 15 ⊢ ∅ ⊆ ◡◡𝑦
67	65, 66	eqsstri 3598	. . . . . . . . . . . . . 14 ⊢ ◡(𝑋 ∖ ◡◡𝑋) ⊆ ◡◡𝑦
68		ssequn2 3748	. . . . . . . . . . . . . 14 ⊢ (◡(𝑋 ∖ ◡◡𝑋) ⊆ ◡◡𝑦 ↔ (◡◡𝑦 ∪ ◡(𝑋 ∖ ◡◡𝑋)) = ◡◡𝑦)
69	67, 68	mpbi 219	. . . . . . . . . . . . 13 ⊢ (◡◡𝑦 ∪ ◡(𝑋 ∖ ◡◡𝑋)) = ◡◡𝑦
70	64, 69	eqtr2i 2633	. . . . . . . . . . . 12 ⊢ ◡◡𝑦 = ◡(◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))
71	63, 70	syl6reqr 2663	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋)) → ◡◡𝑦 = ◡𝑥)
72	71	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → ◡◡𝑦 = ◡𝑥)
73	62, 72	jctild 564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → ((◡𝑋 ⊆ 𝑦 ∧ 𝜒) → (◡◡𝑦 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))))
74	35, 73	spcimedv 3265	. . . . . . . 8 ⊢ (𝜑 → ((◡𝑋 ⊆ 𝑦 ∧ 𝜒) → ∃𝑥(◡◡𝑦 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))))
75	74	imp 444	. . . . . . 7 ⊢ ((𝜑 ∧ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)) → ∃𝑥(◡◡𝑦 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)))
76	75	adantlr 747	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 = ◡◡𝑦) ∧ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)) → ∃𝑥(◡◡𝑦 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)))
77		eqeq1 2614	. . . . . . . . 9 ⊢ (𝑧 = ◡◡𝑦 → (𝑧 = ◡𝑥 ↔ ◡◡𝑦 = ◡𝑥))
78	77	anbi1d 737	. . . . . . . 8 ⊢ (𝑧 = ◡◡𝑦 → ((𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) ↔ (◡◡𝑦 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))))
79	78	exbidv 1837	. . . . . . 7 ⊢ (𝑧 = ◡◡𝑦 → (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) ↔ ∃𝑥(◡◡𝑦 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))))
80	79	ad2antlr 759	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 = ◡◡𝑦) ∧ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)) → (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) ↔ ∃𝑥(◡◡𝑦 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))))
81	76, 80	mpbird 246	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 = ◡◡𝑦) ∧ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)) → ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)))
82	81	ex 449	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ◡◡𝑦) → ((◡𝑋 ⊆ 𝑦 ∧ 𝜒) → ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))))
83		cnvcnvss 5507	. . . . 5 ⊢ ◡◡𝑦 ⊆ 𝑦
84	83	a1i 11	. . . 4 ⊢ (𝜑 → ◡◡𝑦 ⊆ 𝑦)
85	30, 82, 84	intabssd 36935	. . 3 ⊢ (𝜑 → ∩ {𝑧 ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))} ⊆ ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)})
86		vex 3176	. . . . 5 ⊢ 𝑧 ∈ V
87	86	a1i 11	. . . 4 ⊢ (𝜑 → 𝑧 ∈ V)
88		eqtr 2629	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ 𝑧 = ◡𝑥) → 𝑦 = ◡𝑥)
89		cnvss 5216	. . . . . . . . . . . 12 ⊢ (𝑋 ⊆ 𝑥 → ◡𝑋 ⊆ ◡𝑥)
90		sseq2 3590	. . . . . . . . . . . 12 ⊢ (𝑦 = ◡𝑥 → (◡𝑋 ⊆ 𝑦 ↔ ◡𝑋 ⊆ ◡𝑥))
91	89, 90	syl5ibr 235	. . . . . . . . . . 11 ⊢ (𝑦 = ◡𝑥 → (𝑋 ⊆ 𝑥 → ◡𝑋 ⊆ 𝑦))
92	91	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → (𝑋 ⊆ 𝑥 → ◡𝑋 ⊆ 𝑦))
93		clcnvlem.sub2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → (𝜓 → 𝜒))
94	92, 93	anim12d 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → ((𝑋 ⊆ 𝑥 ∧ 𝜓) → (◡𝑋 ⊆ 𝑦 ∧ 𝜒)))
95	94	ex 449	. . . . . . . 8 ⊢ (𝜑 → (𝑦 = ◡𝑥 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) → (◡𝑋 ⊆ 𝑦 ∧ 𝜒))))
96	88, 95	syl5 33	. . . . . . 7 ⊢ (𝜑 → ((𝑦 = 𝑧 ∧ 𝑧 = ◡𝑥) → ((𝑋 ⊆ 𝑥 ∧ 𝜓) → (◡𝑋 ⊆ 𝑦 ∧ 𝜒))))
97	96	impl 648	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝑧) ∧ 𝑧 = ◡𝑥) → ((𝑋 ⊆ 𝑥 ∧ 𝜓) → (◡𝑋 ⊆ 𝑦 ∧ 𝜒)))
98	97	expimpd 627	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) → (◡𝑋 ⊆ 𝑦 ∧ 𝜒)))
99	98	exlimdv 1848	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → (∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓)) → (◡𝑋 ⊆ 𝑦 ∧ 𝜒)))
100		ssid 3587	. . . . 5 ⊢ 𝑧 ⊆ 𝑧
101	100	a1i 11	. . . 4 ⊢ (𝜑 → 𝑧 ⊆ 𝑧)
102	87, 99, 101	intabssd 36935	. . 3 ⊢ (𝜑 → ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)} ⊆ ∩ {𝑧 ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))})
103	85, 102	eqssd 3585	. 2 ⊢ (𝜑 → ∩ {𝑧 ∣ ∃𝑥(𝑧 = ◡𝑥 ∧ (𝑋 ⊆ 𝑥 ∧ 𝜓))} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)})
104	8, 26, 103	3eqtrd 2648	1 ⊢ (𝜑 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∩ cint 4410   × cxp 5036  ^◡ccnv 5037  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060

This theorem is referenced by:  cnvtrucl0  36950  cnvrcl0  36951  cnvtrcl0  36952