Theorem elcnvlem 37907

Description: Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)

Hypothesis

Ref	Expression
elcnvlem.f	|- F = ( x e. ( _V X. _V ) |-> <. ( 2nd ` x ) , ( 1st ` x ) >. )

Assertion

Ref	Expression
elcnvlem	|- ( A e. `' B <-> ( A e. ( _V X. _V ) /\ ( F ` A ) e. B ) )

Proof of Theorem elcnvlem

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

Step	Hyp	Ref	Expression
1		elcnv2 5300	. 2 |- ( A e. `' B <-> E. u E. v ( A = <. u , v >. /\ <. v , u >. e. B ) )
2		fveq2 6191	. . . . 5 |- ( A = <. u , v >. -> ( F ` A ) = ( F ` <. u , v >. ) )
3		vex 3203	. . . . . . 7 |- u e. _V
4		vex 3203	. . . . . . 7 |- v e. _V
5	3, 4	opelvv 5166	. . . . . 6 |- <. u , v >. e. ( _V X. _V )
6	3, 4	op2ndd 7179	. . . . . . . 8 |- ( x = <. u , v >. -> ( 2nd ` x ) = v )
7	3, 4	op1std 7178	. . . . . . . 8 |- ( x = <. u , v >. -> ( 1st ` x ) = u )
8	6, 7	opeq12d 4410	. . . . . . 7 |- ( x = <. u , v >. -> <. ( 2nd ` x ) , ( 1st ` x ) >. = <. v , u >. )
9		elcnvlem.f	. . . . . . 7 |- F = ( x e. ( _V X. _V ) |-> <. ( 2nd ` x ) , ( 1st ` x ) >. )
10		opex 4932	. . . . . . 7 |- <. v , u >. e. _V
11	8, 9, 10	fvmpt 6282	. . . . . 6 |- ( <. u , v >. e. ( _V X. _V ) -> ( F ` <. u , v >. ) = <. v , u >. )
12	5, 11	ax-mp 5	. . . . 5 |- ( F ` <. u , v >. ) = <. v , u >.
13	2, 12	syl6eq 2672	. . . 4 |- ( A = <. u , v >. -> ( F ` A ) = <. v , u >. )
14	13	eleq1d 2686	. . 3 |- ( A = <. u , v >. -> ( ( F ` A ) e. B <-> <. v , u >. e. B ) )
15	14	copsex2gb 5230	. 2 |- ( E. u E. v ( A = <. u , v >. /\ <. v , u >. e. B ) <-> ( A e. ( _V X. _V ) /\ ( F ` A ) e. B ) )
16	1, 15	bitri 264	1 |- ( A e. `' B <-> ( A e. ( _V X. _V ) /\ ( F ` A ) e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  elcnvintab  37908