Theorem ercpbllemg 12749

Description: Lemma for ercpbl 12750. (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 12748	. . 3 |- ( ph -> ( F ` A ) = [ A ] .~ )
6		ercpbllem.b	. . . 4 |- ( ph -> B e. V )
7	1, 2, 3, 6	divsfvalg 12748	. . 3 |- ( ph -> ( F ` B ) = [ B ] .~ )
8	5, 7	eqeq12d 2192	. 2 |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> [ A ] .~ = [ B ] .~ ) )
9	1, 4	erth 6579	. 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 1353   e. wcel 2148   class class class wbr 4004   |-> cmpt 4065   ` cfv 5217   Er wer 6532   [cec 6533

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fv 5225  df-er 6535  df-ec 6537

This theorem is referenced by:  ercpbl  12750  erlecpbl  12751