Theorem eqbrrdva 4950

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 4883	. . . 4 |- ( C X. D ) C_ ( _V X. _V )
3	1, 2	sstrdi 3260	. . 3 |- ( ph -> A C_ ( _V X. _V ) )
4		df-rel 4781	. . 3 |- ( Rel A <-> A C_ ( _V X. _V ) )
5	3, 4	sylibr 134	. 2 |- ( ph -> Rel A )
6		eqbrrdva.2	. . . 4 |- ( ph -> B C_ ( C X. D ) )
7	6, 2	sstrdi 3260	. . 3 |- ( ph -> B C_ ( _V X. _V ) )
8		df-rel 4781	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
9	7, 8	sylibr 134	. 2 |- ( ph -> Rel B )
10	1	ssbrd 4173	. . . 4 |- ( ph -> ( x A y -> x ( C X. D ) y ) )
11		brxp 4805	. . . 4 |- ( x ( C X. D ) y <-> ( x e. C /\ y e. D ) )
12	10, 11	imbitrdi 161	. . 3 |- ( ph -> ( x A y -> ( x e. C /\ y e. D ) ) )
13	6	ssbrd 4173	. . . 4 |- ( ph -> ( x B y -> x ( C X. D ) y ) )
14	13, 11	imbitrdi 161	. . 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 1237	. . 3 |- ( ph -> ( ( x e. C /\ y e. D ) -> ( x A y <-> x B y ) ) )
17	12, 14, 16	pm5.21ndd 717	. 2 |- ( ph -> ( x A y <-> x B y ) )
18	5, 9, 17	eqbrrdv 4872	1 |- ( ph -> A = B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4130   X. cxp 4772   Rel wrel 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781

This theorem is used by: (None)