Theorem refssfne 36391

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

Hypotheses

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

Assertion

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

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

Proof of Theorem refssfne

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

Step	Hyp	Ref	Expression
1		refrel 23421	. . . . . . 7 ⊢ Rel Ref
2	1	brrelex2i 5673	. . . . . 6 ⊢ (𝐵Ref𝐴 → 𝐴 ∈ V)
3	2	adantl 481	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴 ∈ V)
4	1	brrelex1i 5672	. . . . . 6 ⊢ (𝐵Ref𝐴 → 𝐵 ∈ V)
5	4	adantl 481	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐵 ∈ V)
6		unexg 7676	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	3, 5, 6	syl2anc 584	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝐴 ∪ 𝐵) ∈ V)
8		ssun2 4129	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
9	8	a1i 11	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
10		ssun1 4128	. . . . . . 7 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
11	10	a1i 11	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
12		eqimss2 3994	. . . . . . . . 9 ⊢ (𝑋 = 𝑌 → 𝑌 ⊆ 𝑋)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝑌 ⊆ 𝑋)
14		ssequn2 4139	. . . . . . . 8 ⊢ (𝑌 ⊆ 𝑋 ↔ (𝑋 ∪ 𝑌) = 𝑋)
15	13, 14	sylib 218	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑋 ∪ 𝑌) = 𝑋)
16	15	eqcomd 2737	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝑋 = (𝑋 ∪ 𝑌))
17		refssfne.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐴
18		refssfne.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐵
19	17, 18	uneq12i 4116	. . . . . . . 8 ⊢ (𝑋 ∪ 𝑌) = (∪ 𝐴 ∪ ∪ 𝐵)
20		uniun 4882	. . . . . . . 8 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
21	19, 20	eqtr4i 2757	. . . . . . 7 ⊢ (𝑋 ∪ 𝑌) = ∪ (𝐴 ∪ 𝐵)
22	17, 21	fness 36382	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ 𝑋 = (𝑋 ∪ 𝑌)) → 𝐴Fne(𝐴 ∪ 𝐵))
23	7, 11, 16, 22	syl3anc 1373	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴Fne(𝐴 ∪ 𝐵))
24		elun 4103	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
25		ssid 3957	. . . . . . . . . . 11 ⊢ 𝑥 ⊆ 𝑥
26		sseq2 3961	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
27	26	rspcev 3577	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
28	25, 27	mpan2 691	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
30		refssex 23424	. . . . . . . . . . 11 ⊢ ((𝐵Ref𝐴 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
31	30	ex 412	. . . . . . . . . 10 ⊢ (𝐵Ref𝐴 → (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
33	29, 32	jaod 859	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
34	24, 33	biimtrid 242	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑥 ∈ (𝐴 ∪ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
35	34	ralrimiv 3123	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
36	21, 17	isref 23422	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ V → ((𝐴 ∪ 𝐵)Ref𝐴 ↔ (𝑋 = (𝑋 ∪ 𝑌) ∧ ∀𝑥 ∈ (𝐴 ∪ 𝐵)∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
37	7, 36	syl 17	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → ((𝐴 ∪ 𝐵)Ref𝐴 ↔ (𝑋 = (𝑋 ∪ 𝑌) ∧ ∀𝑥 ∈ (𝐴 ∪ 𝐵)∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
38	16, 35, 37	mpbir2and 713	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝐴 ∪ 𝐵)Ref𝐴)
39	9, 23, 38	jca32 515	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴Fne(𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵)Ref𝐴)))
40		sseq2 3961	. . . . . 6 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝐵 ⊆ 𝑐 ↔ 𝐵 ⊆ (𝐴 ∪ 𝐵)))
41		breq2 5095	. . . . . . 7 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝐴Fne𝑐 ↔ 𝐴Fne(𝐴 ∪ 𝐵)))
42		breq1 5094	. . . . . . 7 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝑐Ref𝐴 ↔ (𝐴 ∪ 𝐵)Ref𝐴))
43	41, 42	anbi12d 632	. . . . . 6 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → ((𝐴Fne𝑐 ∧ 𝑐Ref𝐴) ↔ (𝐴Fne(𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵)Ref𝐴)))
44	40, 43	anbi12d 632	. . . . 5 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → ((𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴Fne(𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵)Ref𝐴))))
45	44	spcegv 3552	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ V → ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴Fne(𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵)Ref𝐴)) → ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
46	7, 39, 45	sylc 65	. . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))
47	46	ex 412	. 2 ⊢ (𝑋 = 𝑌 → (𝐵Ref𝐴 → ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
48		vex 3440	. . . . . . . 8 ⊢ 𝑐 ∈ V
49	48	ssex 5259	. . . . . . 7 ⊢ (𝐵 ⊆ 𝑐 → 𝐵 ∈ V)
50	49	ad2antrl 728	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵 ∈ V)
51		simprl 770	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵 ⊆ 𝑐)
52		simpl 482	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = 𝑌)
53		eqid 2731	. . . . . . . . . 10 ⊢ ∪ 𝑐 = ∪ 𝑐
54	53, 17	refbas 23423	. . . . . . . . 9 ⊢ (𝑐Ref𝐴 → 𝑋 = ∪ 𝑐)
55	54	adantl 481	. . . . . . . 8 ⊢ ((𝐴Fne𝑐 ∧ 𝑐Ref𝐴) → 𝑋 = ∪ 𝑐)
56	55	ad2antll 729	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = ∪ 𝑐)
57	52, 56	eqtr3d 2768	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑌 = ∪ 𝑐)
58	18, 53	ssref 23425	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝐵 ⊆ 𝑐 ∧ 𝑌 = ∪ 𝑐) → 𝐵Ref𝑐)
59	50, 51, 57, 58	syl3anc 1373	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵Ref𝑐)
60		simprrr 781	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑐Ref𝐴)
61		reftr 23427	. . . . 5 ⊢ ((𝐵Ref𝑐 ∧ 𝑐Ref𝐴) → 𝐵Ref𝐴)
62	59, 60, 61	syl2anc 584	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵Ref𝐴)
63	62	ex 412	. . 3 ⊢ (𝑋 = 𝑌 → ((𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐵Ref𝐴))
64	63	exlimdv 1934	. 2 ⊢ (𝑋 = 𝑌 → (∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐵Ref𝐴))
65	47, 64	impbid 212	1 ⊢ (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∪ cun 3900   ⊆ wss 3902  ∪ cuni 4859   class class class wbr 5091  Refcref 23415  Fnecfne 36369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-topgen 17344  df-ref 23418  df-fne 36370

This theorem is referenced by: (None)