Theorem fnessref 34546

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 34527	. . . . . . 7 ⊢ Rel Fne
2	1	brrelex2i 5644	. . . . . 6 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V)
3	2	adantl 482	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝐵 ∈ V)
4		rabexg 5255	. . . . 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 4013	. . . . . 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 2830	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑋 ↔ 𝑡 ∈ ∪ 𝐴)
10		eluni 4842	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ∪ 𝐴 ↔ ∃𝑧(𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴))
11	9, 10	bitri 274	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝑋 ↔ ∃𝑧(𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴))
12		fnessex 34535	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴Fne𝐵 ∧ 𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧))
13	12	3expia 1120	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴Fne𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧)))
14	13	adantll 711	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧)))
15		sseq2 3947	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑧))
16	15	rspcev 3561	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑧) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
17	16	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ⊆ 𝑧 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
18	17	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (𝑥 ⊆ 𝑧 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
19	18	anim2d 612	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → ((𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧) → (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
20	19	reximdv 3202	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑧) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
21	14, 20	syld 47	. . . . . . . . . . . . . 14 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ 𝑧 ∈ 𝐴) → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
22	21	ex 413	. . . . . . . . . . . . 13 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑧 ∈ 𝐴 → (𝑡 ∈ 𝑧 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))))
23	22	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑡 ∈ 𝑧 → (𝑧 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))))
24	23	impd 411	. . . . . . . . . . 11 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ((𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
25	24	exlimdv 1936	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (∃𝑧(𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
26	11, 25	syl5bi 241	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑡 ∈ 𝑋 → ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
27		elunirab 4855	. . . . . . . . 9 ⊢ (𝑡 ∈ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ ∃𝑥 ∈ 𝐵 (𝑡 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
28	26, 27	syl6ibr 251	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑡 ∈ 𝑋 → 𝑡 ∈ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}))
29	28	ssrdv 3927	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝑋 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
30	6	unissi 4848	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∪ 𝐵
31		simpl 483	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝑋 = 𝑌)
32		fnessref.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐵
33	31, 32	eqtr2di 2795	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∪ 𝐵 = 𝑋)
34	30, 33	sseqtrid 3973	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝑋)
35	29, 34	eqssd 3938	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝑋 = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
36		fnessex 34535	. . . . . . . . . 10 ⊢ ((𝐴Fne𝐵 ∧ 𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧) → ∃𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
37	36	3expb 1119	. . . . . . . . 9 ⊢ ((𝐴Fne𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ∃𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
38	37	adantll 711	. . . . . . . 8 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ∃𝑤 ∈ 𝐵 (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
39		simpl 483	. . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤 ⊆ 𝑦 ↔ 𝑤 ⊆ 𝑧))
42	41	rspcev 3561	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ⊆ 𝑧) → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦)
43	42	expcom 414	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ⊆ 𝑧 → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
44	43	ad2antll 726	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . 12 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
47	40, 46	jcad 513	. . . . . . . . . . 11 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → (𝑤 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦)))
48		sseq1 3946	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥 ⊆ 𝑦 ↔ 𝑤 ⊆ 𝑦))
49	48	rexbidv 3226	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
50	49	elrab 3624	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ (𝑤 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑤 ⊆ 𝑦))
51	47, 50	syl6ibr 251	. . . . . . . . . 10 ⊢ (((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑡 ∈ 𝑧)) → ((𝑤 ∈ 𝐵 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → 𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}))
52		simpr 485	. . . . . . . . . . 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 513	. . . . . . . . 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 3199	. . . . . . . 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 3123	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ 𝑧 ∃𝑤 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} (𝑡 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
58		eqid 2738	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∪ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}
59	8, 58	isfne2 34531	. . . . . . 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 710	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → 𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
62		sseq1 3946	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑦))
63	62	rexbidv 3226	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦))
64	63	elrab 3624	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦))
65		sseq2 3947	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑤))
66	65	cbvrexvw 3384	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
67	66	biimpi 215	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ⊆ 𝑦 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
68	67	adantl 482	. . . . . . . . 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 241	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
71	70	ralrimiv 3102	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
72	58, 8	isref 22660	. . . . . . 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 710	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴)
75	7, 61, 74	jca32 516	. . . 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 3946	. . . . . 6 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → (𝑐 ⊆ 𝐵 ↔ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ 𝐵))
77		breq2 5078	. . . . . . 7 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → (𝐴Fne𝑐 ↔ 𝐴Fne{𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}))
78		breq1 5077	. . . . . . 7 ⊢ (𝑐 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} → (𝑐Ref𝐴 ↔ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}Ref𝐴))
79	77, 78	anbi12d 631	. . . . . 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 631	. . . . 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 3536	. . . 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 65	. . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴Fne𝐵) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))
83	82	ex 413	. 2 ⊢ (𝑋 = 𝑌 → (𝐴Fne𝐵 → ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
84		simprrl 778	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐴Fne𝑐)
85		eqid 2738	. . . . . . . . . . . 12 ⊢ ∪ 𝑐 = ∪ 𝑐
86	8, 85	fnebas 34533	. . . . . . . . . . 11 ⊢ (𝐴Fne𝑐 → 𝑋 = ∪ 𝑐)
87	84, 86	syl 17	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = ∪ 𝑐)
88		simpl 483	. . . . . . . . . 10 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = 𝑌)
89	87, 88	eqtr3d 2780	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → ∪ 𝑐 = 𝑌)
90	89, 32	eqtrdi 2794	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → ∪ 𝑐 = ∪ 𝐵)
91		vuniex 7592	. . . . . . . 8 ⊢ ∪ 𝑐 ∈ V
92	90, 91	eqeltrrdi 2848	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → ∪ 𝐵 ∈ V)
93		uniexb 7614	. . . . . . 7 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
94	92, 93	sylibr 233	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵 ∈ V)
95		simprl 768	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑐 ⊆ 𝐵)
96	85, 32	fness 34538	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝑐 ⊆ 𝐵 ∧ ∪ 𝑐 = 𝑌) → 𝑐Fne𝐵)
97	94, 95, 89, 96	syl3anc 1370	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑐Fne𝐵)
98		fnetr 34540	. . . . 5 ⊢ ((𝐴Fne𝑐 ∧ 𝑐Fne𝐵) → 𝐴Fne𝐵)
99	84, 97, 98	syl2anc 584	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ (𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐴Fne𝐵)
100	99	ex 413	. . 3 ⊢ (𝑋 = 𝑌 → ((𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐴Fne𝐵))
101	100	exlimdv 1936	. 2 ⊢ (𝑋 = 𝑌 → (∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐴Fne𝐵))
102	83, 101	impbid 211	1 ⊢ (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  Refcref 22653  Fnecfne 34525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-topgen 17154  df-ref 22656  df-fne 34526

This theorem is referenced by: (None)