Theorem eqbrrdva 4782

Description: Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Hypotheses

Ref	Expression
eqbrrdva.1	|- ( ph -> A C_ ( C X. D ) )
eqbrrdva.2	|- ( ph -> B C_ ( C X. D ) )
eqbrrdva.3	|- ( ( ph /\ x e. C /\ y e. D ) -> ( x A y <-> x B y ) )

Assertion

Ref	Expression
eqbrrdva	|- ( ph -> A = B )

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

Allowed substitution hints:   C( x, y)   D( x, y)

Proof of Theorem eqbrrdva

Step	Hyp	Ref	Expression
1		eqbrrdva.1	. . . 4 |- ( ph -> A C_ ( C X. D ) )
2		xpss 4720	. . . 4 |- ( C X. D ) C_ ( _V X. _V )
3	1, 2	sstrdi 3160	. . 3 |- ( ph -> A C_ ( _V X. _V ) )
4		df-rel 4619	. . 3 |- ( Rel A <-> A C_ ( _V X. _V ) )
5	3, 4	sylibr 133	. 2 |- ( ph -> Rel A )
6		eqbrrdva.2	. . . 4 |- ( ph -> B C_ ( C X. D ) )
7	6, 2	sstrdi 3160	. . 3 |- ( ph -> B C_ ( _V X. _V ) )
8		df-rel 4619	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
9	7, 8	sylibr 133	. 2 |- ( ph -> Rel B )
10	1	ssbrd 4033	. . . 4 |- ( ph -> ( x A y -> x ( C X. D ) y ) )
11		brxp 4643	. . . 4 |- ( x ( C X. D ) y <-> ( x e. C /\ y e. D ) )
12	10, 11	syl6ib 160	. . 3 |- ( ph -> ( x A y -> ( x e. C /\ y e. D ) ) )
13	6	ssbrd 4033	. . . 4 |- ( ph -> ( x B y -> x ( C X. D ) y ) )
14	13, 11	syl6ib 160	. . 3 |- ( ph -> ( x B y -> ( x e. C /\ y e. D ) ) )
15		eqbrrdva.3	. . . 4 |- ( ( ph /\ x e. C /\ y e. D ) -> ( x A y <-> x B y ) )
16	15	3expib 1202	. . 3 |- ( ph -> ( ( x e. C /\ y e. D ) -> ( x A y <-> x B y ) ) )
17	12, 14, 16	pm5.21ndd 701	. 2 |- ( ph -> ( x A y <-> x B y ) )
18	5, 9, 17	eqbrrdv 4709	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 974   = wceq 1349   e. wcel 2142   _Vcvv 2731   C_ wss 3122   class class class wbr 3990   X. cxp 4610   Rel wrel 4617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-br 3991  df-opab 4052  df-xp 4618  df-rel 4619

This theorem is referenced by: (None)