Theorem fnessref 31522

Description: A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Hypotheses

Ref	Expression
fnessref.1	⊢ 𝑋 = ∪ 𝐴
fnessref.2	⊢ 𝑌 = ∪ 𝐵

Assertion

Ref	Expression
fnessref	⊢ (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))

Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝑋,𝑐   𝑌,𝑐

Proof of Theorem fnessref

Dummy variables 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnerel 31503	. . . . . . 7 ⊢ Rel Fne
2	1	brrelex2i 5083	. . . . . 6 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V)
3	2	adantl 481	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝐵 ∈ V)
4		rabexg 4739	. . . . 5 ⊢ (𝐵 ∈ V → {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V)
5	3, 4	syl 17	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V)
6		ssrab2 3650	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝐵
7	6	a1i 11	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝐵)
8		fnessref.1	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐴
9	8	eleq2i 2680	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑋 ↔ 𝑡 ∈ ∪ 𝐴)
10		eluni 4375	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ∪ 𝐴 ↔ ∃𝑧(𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴))
11	9, 10	bitri 263	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝑋 ↔ ∃𝑧(𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴))
12		fnessex 31511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴Fne𝐵 ∧ 𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧))
13	12	3expia 1259	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴Fne𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧)))
14	13	adantll 746	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧)))
15		sseq2 3590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑧))
16	15	rspcev 3282	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑧) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
17	16	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ⊆ 𝑧 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
18	17	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (𝑥 ⊆ 𝑧 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
19	18	anim2d 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → ((𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧) → (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
20	19	reximdv 2999	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
21	14, 20	syld 46	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
22	21	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑧 ∈ 𝐴 → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))))
23	22	com23 84	. . . . . . . . . . . 12 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑡 ∈ 𝑧 → (𝑧 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))))
24	23	impd 446	. . . . . . . . . . 11 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ((𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
25	24	exlimdv 1848	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (∃𝑧(𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
26	11, 25	syl5bi 231	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑡 ∈ 𝑋 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
27		elunirab 4384	. . . . . . . . 9 ⊢ (𝑡 ∈ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
28	26, 27	syl6ibr 241	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑡 ∈ 𝑋 → 𝑡 ∈ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}))
29	28	ssrdv 3574	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝑋 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
30	6	unissi 4397	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∪ 𝐵
31		simpl 472	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝑋 = 𝑌)
32		fnessref.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐵
33	31, 32	syl6req 2661	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∪ 𝐵 = 𝑋)
34	30, 33	syl5sseq 3616	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝑋)
35	29, 34	eqssd 3585	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝑋 = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
36		fnessex 31511	. . . . . . . . . 10 ⊢ ((𝐴Fne𝐵 ∧ 𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧) → ∃𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
37	36	3expb 1258	. . . . . . . . 9 ⊢ ((𝐴Fne𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ∃𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
38	37	adantll 746	. . . . . . . 8 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ∃𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
39		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → 𝑤 ∈ 𝐵)
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → 𝑤 ∈ 𝐵))
41		sseq2 3590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤 ⊆ 𝑦 ↔ 𝑤 ⊆ 𝑧))
42	41	rspcev 3282	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ⊆ 𝑧) → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦)
43	42	expcom 450	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ⊆ 𝑧 → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
44	43	ad2antll 761	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
45	44	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
46	45	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
47	40, 46	jcad 554	. . . . . . . . . . 11 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → (𝑤 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦)))
48		sseq1 3589	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥 ⊆ 𝑦 ↔ 𝑤 ⊆ 𝑦))
49	48	rexbidv 3034	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
50	49	elrab 3331	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ (𝑤 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
51	47, 50	syl6ibr 241	. . . . . . . . . 10 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → 𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}))
52		simpr 476	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
53	52	a1i 11	. . . . . . . . . 10 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
54	51, 53	jcad 554	. . . . . . . . 9 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → (𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
55	54	reximdv2 2997	. . . . . . . 8 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → (∃𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) → ∃𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
56	38, 55	mpd 15	. . . . . . 7 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ∃𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
57	56	ralrimivva 2954	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ 𝑧 ∃𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
58		eqid 2610	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}
59	8, 58	isfne2 31507	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V → (𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ (𝑋 = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ 𝑧 ∃𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
60	3, 4, 59	3syl 18	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ (𝑋 = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ 𝑧 ∃𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
61	35, 57, 60	mpbir2and 959	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
62		sseq1 3589	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑦))
63	62	rexbidv 3034	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦))
64	63	elrab 3331	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦))
65		sseq2 3590	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑤))
66	65	cbvrexv 3148	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
67	66	biimpi 205	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
68	67	adantl 481	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
69	68	a1i 11	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ((𝑧 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
70	64, 69	syl5bi 231	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
71	70	ralrimiv 2948	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
72	58, 8	isref 21122	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V → ({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴 ↔ (𝑋 = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
73	3, 4, 72	3syl 18	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴 ↔ (𝑋 = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
74	35, 71, 73	mpbir2and 959	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴)
75	7, 61, 74	jca32 556	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴)))
76		sseq1 3589	. . . . . 6 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → (𝑐 ⊆ 𝐵 ↔ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝐵))
77		breq2 4587	. . . . . . 7 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → (𝐴Fne𝑐 ↔ 𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}))
78		breq1 4586	. . . . . . 7 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → (𝑐Ref𝐴 ↔ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴))
79	77, 78	anbi12d 743	. . . . . 6 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → ((𝐴Fne𝑐 ∧ 𝑐Ref𝐴) ↔ (𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴)))
80	76, 79	anbi12d 743	. . . . 5 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → ((𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) ↔ ({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴))))
81	80	spcegv 3267	. . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V → (({𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∧ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴)) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
82	5, 75, 81	sylc 63	. . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))
83	82	ex 449	. 2 ⊢ (𝑋 = 𝑌 → (𝐴Fne𝐵 → ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
84		simprrl 800	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐴Fne𝑐)
85		eqid 2610	. . . . . . . . . . . 12 ⊢ ∪ 𝑐 = ∪ 𝑐
86	8, 85	fnebas 31509	. . . . . . . . . . 11 ⊢ (𝐴Fne𝑐 → 𝑋 = ∪ 𝑐)
87	84, 86	syl 17	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = ∪ 𝑐)
88		simpl 472	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = 𝑌)
89	87, 88	eqtr3d 2646	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → ∪ 𝑐 = 𝑌)
90	89, 32	syl6eq 2660	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → ∪ 𝑐 = ∪ 𝐵)
91		vuniex 6852	. . . . . . . 8 ⊢ ∪ 𝑐 ∈ V
92	90, 91	syl6eqelr 2697	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → ∪ 𝐵 ∈ V)
93		uniexb 6866	. . . . . . 7 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
94	92, 93	sylibr 223	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵 ∈ V)
95		simprl 790	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑐 ⊆ 𝐵)
96	85, 32	fness 31514	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑐 ⊆ 𝐵 ∧ ∪ 𝑐 = 𝑌) → 𝑐Fne𝐵)
97	94, 95, 89, 96	syl3anc 1318	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑐Fne𝐵)
98		fnetr 31516	. . . . 5 ⊢ ((𝐴Fne𝑐 ∧ 𝑐Fne𝐵) → 𝐴Fne𝐵)
99	84, 97, 98	syl2anc 691	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐴Fne𝐵)
100	99	ex 449	. . 3 ⊢ (𝑋 = 𝑌 → ((𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐴Fne𝐵))
101	100	exlimdv 1848	. 2 ⊢ (𝑋 = 𝑌 → (∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐴Fne𝐵))
102	83, 101	impbid 201	1 ⊢ (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  Refcref 21115  Fnecfne 31501

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-ne 2782  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-iun 4457  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-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927  df-ref 21118  df-fne 31502

This theorem is referenced by: (None)