Theorem eqbrrdva 4848

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 4783	. . . 4 |- ( C X. D ) C_ ( _V X. _V )
3	1, 2	sstrdi 3205	. . 3 |- ( ph -> A C_ ( _V X. _V ) )
4		df-rel 4682	. . 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 3205	. . 3 |- ( ph -> B C_ ( _V X. _V ) )
8		df-rel 4682	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
9	7, 8	sylibr 134	. 2 |- ( ph -> Rel B )
10	1	ssbrd 4087	. . . 4 |- ( ph -> ( x A y -> x ( C X. D ) y ) )
11		brxp 4706	. . . 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 4087	. . . 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 1209	. . 3 |- ( ph -> ( ( x e. C /\ y e. D ) -> ( x A y <-> x B y ) ) )
17	12, 14, 16	pm5.21ndd 707	. 2 |- ( ph -> ( x A y <-> x B y ) )
18	5, 9, 17	eqbrrdv 4772	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   class class class wbr 4044   X. cxp 4673   Rel wrel 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682

This theorem is referenced by: (None)