Theorem divsfvalg 12753

Description: Value of the function in qusval 12749. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-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 ] .~ )
ercpbl.a	|- ( ph -> A e. V )

Assertion

Ref	Expression
divsfvalg	|- ( ph -> ( F ` A ) = [ A ] .~ )

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

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

Proof of Theorem divsfvalg

Step	Hyp	Ref	Expression
1		ercpbl.f	. 2 |- F = ( x e. V |-> [ x ] .~ )
2		eceq1 6572	. 2 |- ( x = A -> [ x ] .~ = [ A ] .~ )
3		ercpbl.a	. 2 |- ( ph -> A e. V )
4		ercpbl.v	. . 3 |- ( ph -> V e. W )
5		ercpbl.r	. . . 4 |- ( ph -> .~ Er V )
6	5	ecss 6578	. . 3 |- ( ph -> [ A ] .~ C_ V )
7	4, 6	ssexd 4145	. 2 |- ( ph -> [ A ] .~ e. _V )
8	1, 2, 3, 7	fvmptd3 5611	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   |-> cmpt 4066   ` cfv 5218   Er wer 6534   [cec 6535

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 4123  ax-pow 4176  ax-pr 4211

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 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fv 5226  df-er 6537  df-ec 6539

This theorem is referenced by:  ercpbllemg  12754  qusaddvallemg  12757