Theorem refssfne 32353

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 21311	. . . . . . 7 ⊢ Rel Ref
2	1	brrelex2i 5159	. . . . . 6 ⊢ (𝐵Ref𝐴 → 𝐴 ∈ V)
3	2	adantl 482	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴 ∈ V)
4	1	brrelexi 5158	. . . . . 6 ⊢ (𝐵Ref𝐴 → 𝐵 ∈ V)
5	4	adantl 482	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐵 ∈ V)
6		unexg 6959	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	3, 5, 6	syl2anc 693	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝐴 ∪ 𝐵) ∈ V)
8		ssun2 3777	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
9	8	a1i 11	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
10		ssun1 3776	. . . . . . 7 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
11	10	a1i 11	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
12		eqimss2 3658	. . . . . . . . 9 ⊢ (𝑋 = 𝑌 → 𝑌 ⊆ 𝑋)
13	12	adantr 481	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝑌 ⊆ 𝑋)
14		ssequn2 3786	. . . . . . . 8 ⊢ (𝑌 ⊆ 𝑋 ↔ (𝑋 ∪ 𝑌) = 𝑋)
15	13, 14	sylib 208	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑋 ∪ 𝑌) = 𝑋)
16	15	eqcomd 2628	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝑋 = (𝑋 ∪ 𝑌))
17		refssfne.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐴
18		refssfne.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐵
19	17, 18	uneq12i 3765	. . . . . . . 8 ⊢ (𝑋 ∪ 𝑌) = (∪ 𝐴 ∪ ∪ 𝐵)
20		uniun 4456	. . . . . . . 8 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
21	19, 20	eqtr4i 2647	. . . . . . 7 ⊢ (𝑋 ∪ 𝑌) = ∪ (𝐴 ∪ 𝐵)
22	17, 21	fness 32344	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ 𝑋 = (𝑋 ∪ 𝑌)) → 𝐴Fne(𝐴 ∪ 𝐵))
23	7, 11, 16, 22	syl3anc 1326	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → 𝐴Fne(𝐴 ∪ 𝐵))
24		elun 3753	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
25		ssid 3624	. . . . . . . . . . 11 ⊢ 𝑥 ⊆ 𝑥
26		sseq2 3627	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
27	26	rspcev 3309	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
28	25, 27	mpan2 707	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
30		refssex 21314	. . . . . . . . . . 11 ⊢ ((𝐵Ref𝐴 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
31	30	ex 450	. . . . . . . . . 10 ⊢ (𝐵Ref𝐴 → (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
32	31	adantl 482	. . . . . . . . 9 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
33	29, 32	jaod 395	. . . . . . . 8 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
34	24, 33	syl5bi 232	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝑥 ∈ (𝐴 ∪ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))
35	34	ralrimiv 2965	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
36	21, 17	isref 21312	. . . . . . 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 957	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝐴 ∪ 𝐵)Ref𝐴)
39	9, 23, 38	jca32 558	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐵Ref𝐴) → (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴Fne(𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵)Ref𝐴)))
40		sseq2 3627	. . . . . 6 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝐵 ⊆ 𝑐 ↔ 𝐵 ⊆ (𝐴 ∪ 𝐵)))
41		breq2 4657	. . . . . . 7 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝐴Fne𝑐 ↔ 𝐴Fne(𝐴 ∪ 𝐵)))
42		breq1 4656	. . . . . . 7 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → (𝑐Ref𝐴 ↔ (𝐴 ∪ 𝐵)Ref𝐴))
43	41, 42	anbi12d 747	. . . . . 6 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → ((𝐴Fne𝑐 ∧ 𝑐Ref𝐴) ↔ (𝐴Fne(𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵)Ref𝐴)))
44	40, 43	anbi12d 747	. . . . 5 ⊢ (𝑐 = (𝐴 ∪ 𝐵) → ((𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴Fne(𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵)Ref𝐴))))
45	44	spcegv 3294	. . . 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 450	. 2 ⊢ (𝑋 = 𝑌 → (𝐵Ref𝐴 → ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
48		vex 3203	. . . . . . . 8 ⊢ 𝑐 ∈ V
49	48	ssex 4802	. . . . . . 7 ⊢ (𝐵 ⊆ 𝑐 → 𝐵 ∈ V)
50	49	ad2antrl 764	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵 ∈ V)
51		simprl 794	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵 ⊆ 𝑐)
52		simpl 473	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = 𝑌)
53		eqid 2622	. . . . . . . . . 10 ⊢ ∪ 𝑐 = ∪ 𝑐
54	53, 17	refbas 21313	. . . . . . . . 9 ⊢ (𝑐Ref𝐴 → 𝑋 = ∪ 𝑐)
55	54	adantl 482	. . . . . . . 8 ⊢ ((𝐴Fne𝑐 ∧ 𝑐Ref𝐴) → 𝑋 = ∪ 𝑐)
56	55	ad2antll 765	. . . . . . 7 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑋 = ∪ 𝑐)
57	52, 56	eqtr3d 2658	. . . . . 6 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑌 = ∪ 𝑐)
58	18, 53	ssref 21315	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝐵 ⊆ 𝑐 ∧ 𝑌 = ∪ 𝑐) → 𝐵Ref𝑐)
59	50, 51, 57, 58	syl3anc 1326	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵Ref𝑐)
60		simprrr 805	. . . . 5 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝑐Ref𝐴)
61		reftr 21317	. . . . 5 ⊢ ((𝐵Ref𝑐 ∧ 𝑐Ref𝐴) → 𝐵Ref𝐴)
62	59, 60, 61	syl2anc 693	. . . 4 ⊢ ((𝑋 = 𝑌 ∧ (𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))) → 𝐵Ref𝐴)
63	62	ex 450	. . 3 ⊢ (𝑋 = 𝑌 → ((𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐵Ref𝐴))
64	63	exlimdv 1861	. 2 ⊢ (𝑋 = 𝑌 → (∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)) → 𝐵Ref𝐴))
65	47, 64	impbid 202	1 ⊢ (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∪ cuni 4436   class class class wbr 4653  Refcref 21305  Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-ref 21308  df-fne 32332

This theorem is referenced by: (None)