Theorem divsfvalg 13593

Description: Value of the function in qusval 13587. (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 6815	. 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 6823	. . 3 |- ( ph -> [ A ] .~ C_ V )
7	4, 6	ssexd 4255	. 2 |- ( ph -> [ A ] .~ e. _V )
8	1, 2, 3, 7	fvmptd3 5776	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   |-> cmpt 4176   ` cfv 5357   Er wer 6777   [cec 6778

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fv 5365  df-er 6780  df-ec 6782

This theorem is referenced by:  ercpbllemg  13594  qusaddvallemg  13597  qusgrp2  13866  qusring2  14309