Theorem divsfvalg 13357

Description: Value of the function in qusval 13351. (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 6713	. 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 6721	. . 3 |- ( ph -> [ A ] .~ C_ V )
7	4, 6	ssexd 4223	. 2 |- ( ph -> [ A ] .~ e. _V )
8	1, 2, 3, 7	fvmptd3 5727	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   |-> cmpt 4144   ` cfv 5317   Er wer 6675   [cec 6676

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fv 5325  df-er 6678  df-ec 6680

This theorem is referenced by:  ercpbllemg  13358  qusaddvallemg  13361  qusgrp2  13645  qusring2  14024