Theorem ercpbllemg 13277

Description: Lemma for ercpbl 13278. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
ercpbl.r	|- ( ph -> .~ Er V )
ercpbl.v	|- ( ph -> V e. W )
ercpbl.f	|- F = ( x e. V |-> [ x ] .~ )
ercpbllem.1	|- ( ph -> A e. V )
ercpbllem.b	|- ( ph -> B e. V )

Assertion

Ref	Expression
ercpbllemg	|- ( ph -> ( ( F ` A ) = ( F ` B ) <-> A .~ B ) )

Distinct variable groups:   x, .~   x, A   x, B   x, V   ph, x

Allowed substitution hints:   F( x)   W( x)

Proof of Theorem ercpbllemg

Step	Hyp	Ref	Expression
1		ercpbl.r	. . . 4 |- ( ph -> .~ Er V )
2		ercpbl.v	. . . 4 |- ( ph -> V e. W )
3		ercpbl.f	. . . 4 |- F = ( x e. V |-> [ x ] .~ )
4		ercpbllem.1	. . . 4 |- ( ph -> A e. V )
5	1, 2, 3, 4	divsfvalg 13276	. . 3 |- ( ph -> ( F ` A ) = [ A ] .~ )
6		ercpbllem.b	. . . 4 |- ( ph -> B e. V )
7	1, 2, 3, 6	divsfvalg 13276	. . 3 |- ( ph -> ( F ` B ) = [ B ] .~ )
8	5, 7	eqeq12d 2222	. 2 |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> [ A ] .~ = [ B ] .~ ) )
9	1, 4	erth 6689	. 2 |- ( ph -> ( A .~ B <-> [ A ] .~ = [ B ] .~ ) )
10	8, 9	bitr4d 191	1 |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> A .~ B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   class class class wbr 4059   |-> cmpt 4121   ` cfv 5290   Er wer 6640   [cec 6641

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fv 5298  df-er 6643  df-ec 6645

This theorem is referenced by:  ercpbl  13278  erlecpbl  13279