Theorem ercpbllemg 13628

Description: Lemma for ercpbl 13629. (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 13627	. . 3 |- ( ph -> ( F ` A ) = [ A ] .~ )
6		ercpbllem.b	. . . 4 |- ( ph -> B e. V )
7	1, 2, 3, 6	divsfvalg 13627	. . 3 |- ( ph -> ( F ` B ) = [ B ] .~ )
8	5, 7	eqeq12d 2253	. 2 |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> [ A ] .~ = [ B ] .~ ) )
9	1, 4	erth 6843	. 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 1402   e. wcel 2209   class class class wbr 4125   |-> cmpt 4187   ` cfv 5372   Er wer 6794   [cec 6795

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fv 5380  df-er 6797  df-ec 6799

This theorem is referenced by:  ercpbl  13629  erlecpbl  13630