Theorem isref 21312

Description: The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 32334. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Hypotheses

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

Assertion

Ref	Expression
isref	|- ( A e. C -> ( A Ref B <-> ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) )

Distinct variable groups:   x, A   x, y, B

Allowed substitution hints:   A( y)   C( x, y)   X( x, y)   Y( x, y)

Proof of Theorem isref

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		refrel 21311	. . . 4 |- Rel Ref
2	1	brrelex2i 5159	. . 3 |- ( A Ref B -> B e. _V )
3	2	anim2i 593	. 2 |- ( ( A e. C /\ A Ref B ) -> ( A e. C /\ B e. _V ) )
4		simpl 473	. . 3 |- ( ( A e. C /\ ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) -> A e. C )
5		simpr 477	. . . . . . 7 |- ( ( A e. C /\ Y = X ) -> Y = X )
6		isref.2	. . . . . . 7 |- Y = U. B
7		isref.1	. . . . . . 7 |- X = U. A
8	5, 6, 7	3eqtr3g 2679	. . . . . 6 |- ( ( A e. C /\ Y = X ) -> U. B = U. A )
9		uniexg 6955	. . . . . . 7 |- ( A e. C -> U. A e. _V )
10	9	adantr 481	. . . . . 6 |- ( ( A e. C /\ Y = X ) -> U. A e. _V )
11	8, 10	eqeltrd 2701	. . . . 5 |- ( ( A e. C /\ Y = X ) -> U. B e. _V )
12		uniexb 6973	. . . . 5 |- ( B e. _V <-> U. B e. _V )
13	11, 12	sylibr 224	. . . 4 |- ( ( A e. C /\ Y = X ) -> B e. _V )
14	13	adantrr 753	. . 3 |- ( ( A e. C /\ ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) -> B e. _V )
15	4, 14	jca 554	. 2 |- ( ( A e. C /\ ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) -> ( A e. C /\ B e. _V ) )
16		unieq 4444	. . . . . 6 |- ( a = A -> U. a = U. A )
17	16, 7	syl6eqr 2674	. . . . 5 |- ( a = A -> U. a = X )
18	17	eqeq2d 2632	. . . 4 |- ( a = A -> ( U. b = U. a <-> U. b = X ) )
19		raleq 3138	. . . 4 |- ( a = A -> ( A. x e. a E. y e. b x C_ y <-> A. x e. A E. y e. b x C_ y ) )
20	18, 19	anbi12d 747	. . 3 |- ( a = A -> ( ( U. b = U. a /\ A. x e. a E. y e. b x C_ y ) <-> ( U. b = X /\ A. x e. A E. y e. b x C_ y ) ) )
21		unieq 4444	. . . . . 6 |- ( b = B -> U. b = U. B )
22	21, 6	syl6eqr 2674	. . . . 5 |- ( b = B -> U. b = Y )
23	22	eqeq1d 2624	. . . 4 |- ( b = B -> ( U. b = X <-> Y = X ) )
24		rexeq 3139	. . . . 5 |- ( b = B -> ( E. y e. b x C_ y <-> E. y e. B x C_ y ) )
25	24	ralbidv 2986	. . . 4 |- ( b = B -> ( A. x e. A E. y e. b x C_ y <-> A. x e. A E. y e. B x C_ y ) )
26	23, 25	anbi12d 747	. . 3 |- ( b = B -> ( ( U. b = X /\ A. x e. A E. y e. b x C_ y ) <-> ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) )
27		df-ref 21308	. . 3 |- Ref = { <. a , b >. | ( U. b = U. a /\ A. x e. a E. y e. b x C_ y ) }
28	20, 26, 27	brabg 4994	. 2 |- ( ( A e. C /\ B e. _V ) -> ( A Ref B <-> ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) )
29	3, 15, 28	pm5.21nd 941	1 |- ( A e. C -> ( A Ref B <-> ( Y = X /\ A. x e. A E. y e. B x C_ y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   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:  refbas  21313  refssex  21314  ssref  21315  refref  21316  reftr  21317  refun0  21318  dissnref  21331  reff  29906  locfinreflem  29907  cmpcref  29917  fnessref  32352  refssfne  32353