Theorem eqbrrdva 4774

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 4712	. . . 4 |- ( C X. D ) C_ ( _V X. _V )
3	1, 2	sstrdi 3154	. . 3 |- ( ph -> A C_ ( _V X. _V ) )
4		df-rel 4611	. . 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 3154	. . 3 |- ( ph -> B C_ ( _V X. _V ) )
8		df-rel 4611	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
9	7, 8	sylibr 133	. 2 |- ( ph -> Rel B )
10	1	ssbrd 4025	. . . 4 |- ( ph -> ( x A y -> x ( C X. D ) y ) )
11		brxp 4635	. . . 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 4025	. . . 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 1196	. . 3 |- ( ph -> ( ( x e. C /\ y e. D ) -> ( x A y <-> x B y ) ) )
17	12, 14, 16	pm5.21ndd 695	. 2 |- ( ph -> ( x A y <-> x B y ) )
18	5, 9, 17	eqbrrdv 4701	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   class class class wbr 3982   X. cxp 4602   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by: (None)