Theorem ercpbllemg 12913

Description: Lemma for ercpbl 12914. (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 12912	. . 3 |- ( ph -> ( F ` A ) = [ A ] .~ )
6		ercpbllem.b	. . . 4 |- ( ph -> B e. V )
7	1, 2, 3, 6	divsfvalg 12912	. . 3 |- ( ph -> ( F ` B ) = [ B ] .~ )
8	5, 7	eqeq12d 2208	. 2 |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> [ A ] .~ = [ B ] .~ ) )
9	1, 4	erth 6633	. 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 1364   e. wcel 2164   class class class wbr 4029   |-> cmpt 4090   ` cfv 5254   Er wer 6584   [cec 6585

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fv 5262  df-er 6587  df-ec 6589

This theorem is referenced by:  ercpbl  12914  erlecpbl  12915