Theorem divsfvalg 13161

Description: Value of the function in qusval 13155. (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 6655	. 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 6663	. . 3 |- ( ph -> [ A ] .~ C_ V )
7	4, 6	ssexd 4184	. 2 |- ( ph -> [ A ] .~ e. _V )
8	1, 2, 3, 7	fvmptd3 5673	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   |-> cmpt 4105   ` cfv 5271   Er wer 6617   [cec 6618

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fv 5279  df-er 6620  df-ec 6622

This theorem is referenced by:  ercpbllemg  13162  qusaddvallemg  13165  qusgrp2  13449  qusring2  13828