Theorem refssfne 36353

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 23402	. . . . . . 7 ⊢ Rel Ref
2	1	brrelex2i 5698	. . . . . 6 ⊢ (𝐵Ref𝐴 → 𝐴 ∈ V)
3	2	adantl 481	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴 ∈ V)
4	1	brrelex1i 5697	. . . . . 6 ⊢ (𝐵Ref𝐴 → 𝐵 ∈ V)
5	4	adantl 481	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐵 ∈ V)
6		unexg 7722	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	3, 5, 6	syl2anc 584	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝐴 ∪ 𝐵) ∈ V)
8		ssun2 4145	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
9	8	a1i 11	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
10		ssun1 4144	. . . . . . 7 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
11	10	a1i 11	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
12		eqimss2 4009	. . . . . . . . 9 ⊢ (𝑋 = 𝑌 → 𝑌 ⊆ 𝑋)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝑌 ⊆ 𝑋)
14		ssequn2 4155	. . . . . . . 8 ⊢ (𝑌 ⊆ 𝑋 ↔ (𝑋 ∪ 𝑌) = 𝑋)
15	13, 14	sylib 218	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑋 ∪ 𝑌) = 𝑋)
16	15	eqcomd 2736	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝑋 = (𝑋 ∪ 𝑌))
17		refssfne.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐴
18		refssfne.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐵
19	17, 18	uneq12i 4132	. . . . . . . 8 ⊢ (𝑋 ∪ 𝑌) = (∪ 𝐴 ∪ ∪ 𝐵)
20		uniun 4897	. . . . . . . 8 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
21	19, 20	eqtr4i 2756	. . . . . . 7 ⊢ (𝑋 ∪ 𝑌) = ∪ (𝐴 ∪ 𝐵)
22	17, 21	fness 36344	. . . . . 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 4119	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
25		ssid 3972	. . . . . . . . . . 11 ⊢ 𝑥 ⊆ 𝑥
26		sseq2 3976	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
27	26	rspcev 3591	. . . . . . . . . . 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 23405	. . . . . . . . . . 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 3125	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
36	21, 17	isref 23403	. . . . . . 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 3976	. . . . . 6 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝐵 ⊆ 𝑐 ↔ 𝐵 ⊆ (𝐴 ∪ 𝐵)))
41		breq2 5114	. . . . . . 7 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝐴Fne𝑐 ↔ 𝐴Fne(𝐴 ∪ 𝐵)))
42		breq1 5113	. . . . . . 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 3566	. . . 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 3454	. . . . . . . 8 ⊢ 𝑐 ∈ V
49	48	ssex 5279	. . . . . . 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 2730	. . . . . . . . . 10 ⊢ ∪ 𝑐 = ∪ 𝑐
54	53, 17	refbas 23404	. . . . . . . . 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 2767	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑌 = ∪ 𝑐)
58	18, 53	ssref 23406	. . . . . 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 23408	. . . . 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 1933	. 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 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  ∪ cuni 4874   class class class wbr 5110  Refcref 23396  Fnecfne 36331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-topgen 17413  df-ref 23399  df-fne 36332

This theorem is referenced by: (None)