Theorem ssref 21315

Description: A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Hypotheses

Ref	Expression
ssref.1	|- X = U. A
ssref.2	|- Y = U. B

Assertion

Ref	Expression
ssref	|- ( ( A e. C /\ A C_ B /\ X = Y ) -> A Ref B )

Proof of Theorem ssref

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

Step	Hyp	Ref	Expression
1		eqcom 2629	. . . 4 |- ( X = Y <-> Y = X )
2	1	biimpi 206	. . 3 |- ( X = Y -> Y = X )
3	2	3ad2ant3 1084	. 2 |- ( ( A e. C /\ A C_ B /\ X = Y ) -> Y = X )
4		ssel2 3598	. . . . 5 |- ( ( A C_ B /\ x e. A ) -> x e. B )
5	4	3ad2antl2 1224	. . . 4 |- ( ( ( A e. C /\ A C_ B /\ X = Y ) /\ x e. A ) -> x e. B )
6		ssid 3624	. . . 4 |- x C_ x
7		sseq2 3627	. . . . 5 |- ( y = x -> ( x C_ y <-> x C_ x ) )
8	7	rspcev 3309	. . . 4 |- ( ( x e. B /\ x C_ x ) -> E. y e. B x C_ y )
9	5, 6, 8	sylancl 694	. . 3 |- ( ( ( A e. C /\ A C_ B /\ X = Y ) /\ x e. A ) -> E. y e. B x C_ y )
10	9	ralrimiva 2966	. 2 |- ( ( A e. C /\ A C_ B /\ X = Y ) -> A. x e. A E. y e. B x C_ y )
11		ssref.1	. . . 4 |- X = U. A
12		ssref.2	. . . 4 |- Y = U. B
13	11, 12	isref 21312	. . 3 |- ( A e. C -> ( A Ref B <-> ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) )
14	13	3ad2ant1 1082	. 2 |- ( ( A e. C /\ A C_ B /\ X = Y ) -> ( A Ref B <-> ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) )
15	3, 10, 14	mpbir2and 957	1 |- ( ( A e. C /\ A C_ B /\ X = Y ) -> A Ref B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   class class class wbr 4653   Refcref 21305

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-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-xp 5120  df-rel 5121  df-ref 21308

This theorem is referenced by:  cmpcref  29917  refssfne  32353