Theorem divsfval 16207

Description: Value of the function in qusval 16202. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ercpbl.r	|- ( ph -> .~ Er V )
ercpbl.v	|- ( ph -> V e. _V )
ercpbl.f	|- F = ( x e. V |-> [ x ] .~ )

Assertion

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

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

Allowed substitution hint:   F( x)

Proof of Theorem divsfval

Step	Hyp	Ref	Expression
1		ercpbl.v	. . . . 5 |- ( ph -> V e. _V )
2		ercpbl.r	. . . . . 6 |- ( ph -> .~ Er V )
3	2	ecss 7788	. . . . 5 |- ( ph -> [ A ] .~ C_ V )
4	1, 3	ssexd 4805	. . . 4 |- ( ph -> [ A ] .~ e. _V )
5		eceq1 7782	. . . . 5 |- ( x = A -> [ x ] .~ = [ A ] .~ )
6		ercpbl.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
7	5, 6	fvmptg 6280	. . . 4 |- ( ( A e. V /\ [ A ] .~ e. _V ) -> ( F ` A ) = [ A ] .~ )
8	4, 7	sylan2 491	. . 3 |- ( ( A e. V /\ ph ) -> ( F ` A ) = [ A ] .~ )
9	8	expcom 451	. 2 |- ( ph -> ( A e. V -> ( F ` A ) = [ A ] .~ ) )
10	6	dmeqi 5325	. . . . . . . 8 |- dom F = dom ( x e. V |-> [ x ] .~ )
11	2	ecss 7788	. . . . . . . . . . 11 |- ( ph -> [ x ] .~ C_ V )
12	1, 11	ssexd 4805	. . . . . . . . . 10 |- ( ph -> [ x ] .~ e. _V )
13	12	ralrimivw 2967	. . . . . . . . 9 |- ( ph -> A. x e. V [ x ] .~ e. _V )
14		dmmptg 5632	. . . . . . . . 9 |- ( A. x e. V [ x ] .~ e. _V -> dom ( x e. V |-> [ x ] .~ ) = V )
15	13, 14	syl 17	. . . . . . . 8 |- ( ph -> dom ( x e. V |-> [ x ] .~ ) = V )
16	10, 15	syl5eq 2668	. . . . . . 7 |- ( ph -> dom F = V )
17	16	eleq2d 2687	. . . . . 6 |- ( ph -> ( A e. dom F <-> A e. V ) )
18	17	notbid 308	. . . . 5 |- ( ph -> ( -. A e. dom F <-> -. A e. V ) )
19		ndmfv 6218	. . . . 5 |- ( -. A e. dom F -> ( F ` A ) = (/) )
20	18, 19	syl6bir 244	. . . 4 |- ( ph -> ( -. A e. V -> ( F ` A ) = (/) ) )
21		ecdmn0 7789	. . . . . 6 |- ( A e. dom .~ <-> [ A ] .~ =/= (/) )
22		erdm 7752	. . . . . . . . 9 |- ( .~ Er V -> dom .~ = V )
23	2, 22	syl 17	. . . . . . . 8 |- ( ph -> dom .~ = V )
24	23	eleq2d 2687	. . . . . . 7 |- ( ph -> ( A e. dom .~ <-> A e. V ) )
25	24	biimpd 219	. . . . . 6 |- ( ph -> ( A e. dom .~ -> A e. V ) )
26	21, 25	syl5bir 233	. . . . 5 |- ( ph -> ( [ A ] .~ =/= (/) -> A e. V ) )
27	26	necon1bd 2812	. . . 4 |- ( ph -> ( -. A e. V -> [ A ] .~ = (/) ) )
28	20, 27	jcad 555	. . 3 |- ( ph -> ( -. A e. V -> ( ( F ` A ) = (/) /\ [ A ] .~ = (/) ) ) )
29		eqtr3 2643	. . 3 |- ( ( ( F ` A ) = (/) /\ [ A ] .~ = (/) ) -> ( F ` A ) = [ A ] .~ )
30	28, 29	syl6 35	. 2 |- ( ph -> ( -. A e. V -> ( F ` A ) = [ A ] .~ ) )
31	9, 30	pm2.61d 170	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ` cfv 5888   Er wer 7739   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-er 7742  df-ec 7744

This theorem is referenced by:  ercpbllem  16208  qusaddvallem  16211  qusgrp2  17533  frgpmhm  18178  frgpup3lem  18190  qusring2  18620  qusrhm  19237