Theorem eqrelf 34020

Description: The equality connective between relations. (Contributed by Peter Mazsa, 25-Jun-2019.)

Hypotheses

Ref	Expression
eqrelf.1	|- F/_ x A
eqrelf.2	|- F/_ x B
eqrelf.3	|- F/_ y A
eqrelf.4	|- F/_ y B

Assertion

Ref	Expression
eqrelf	|- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)

Proof of Theorem eqrelf

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqrel 5209	. 2 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. u A. v ( <. u , v >. e. A <-> <. u , v >. e. B ) ) )
2		nfv 1843	. . 3 |- F/ u ( <. x , y >. e. A <-> <. x , y >. e. B )
3		nfv 1843	. . 3 |- F/ v ( <. x , y >. e. A <-> <. x , y >. e. B )
4		eqrelf.1	. . . . 5 |- F/_ x A
5	4	nfel2 2781	. . . 4 |- F/ x <. u , v >. e. A
6		eqrelf.2	. . . . 5 |- F/_ x B
7	6	nfel2 2781	. . . 4 |- F/ x <. u , v >. e. B
8	5, 7	nfbi 1833	. . 3 |- F/ x ( <. u , v >. e. A <-> <. u , v >. e. B )
9		eqrelf.3	. . . . 5 |- F/_ y A
10	9	nfel2 2781	. . . 4 |- F/ y <. u , v >. e. A
11		eqrelf.4	. . . . 5 |- F/_ y B
12	11	nfel2 2781	. . . 4 |- F/ y <. u , v >. e. B
13	10, 12	nfbi 1833	. . 3 |- F/ y ( <. u , v >. e. A <-> <. u , v >. e. B )
14		opeq12 4404	. . . . 5 |- ( ( x = u /\ y = v ) -> <. x , y >. = <. u , v >. )
15	14	eleq1d 2686	. . . 4 |- ( ( x = u /\ y = v ) -> ( <. x , y >. e. A <-> <. u , v >. e. A ) )
16	14	eleq1d 2686	. . . 4 |- ( ( x = u /\ y = v ) -> ( <. x , y >. e. B <-> <. u , v >. e. B ) )
17	15, 16	bibi12d 335	. . 3 |- ( ( x = u /\ y = v ) -> ( ( <. x , y >. e. A <-> <. x , y >. e. B ) <-> ( <. u , v >. e. A <-> <. u , v >. e. B ) ) )
18	2, 3, 8, 13, 17	cbval2 2279	. 2 |- ( A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) <-> A. u A. v ( <. u , v >. e. A <-> <. u , v >. e. B ) )
19	1, 18	syl6bbr 278	1 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   F/_wnfc 2751   <.cop 4183   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  vvdifopab  34024