Theorem eqbrrdva 4833

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 4768	. . . 4 |- ( C X. D ) C_ ( _V X. _V )
3	1, 2	sstrdi 3192	. . 3 |- ( ph -> A C_ ( _V X. _V ) )
4		df-rel 4667	. . 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 3192	. . 3 |- ( ph -> B C_ ( _V X. _V ) )
8		df-rel 4667	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
9	7, 8	sylibr 134	. 2 |- ( ph -> Rel B )
10	1	ssbrd 4073	. . . 4 |- ( ph -> ( x A y -> x ( C X. D ) y ) )
11		brxp 4691	. . . 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 4073	. . . 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 1208	. . 3 |- ( ph -> ( ( x e. C /\ y e. D ) -> ( x A y <-> x B y ) ) )
17	12, 14, 16	pm5.21ndd 706	. 2 |- ( ph -> ( x A y <-> x B y ) )
18	5, 9, 17	eqbrrdv 4757	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3154   class class class wbr 4030   X. cxp 4658   Rel wrel 4665

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667

This theorem is referenced by: (None)