Theorem divsfval 12911

Description: Value of the function in qusval 12906. (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 ] .~ )

Assertion

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

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

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

Proof of Theorem divsfval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ercpbl.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
2	1	mptrcl 5640	. . . 4 |- ( y e. ( F ` A ) -> A e. V )
3	2	a1i 9	. . 3 |- ( ph -> ( y e. ( F ` A ) -> A e. V ) )
4		19.8a 1601	. . . 4 |- ( y e. [ A ] .~ -> E. y y e. [ A ] .~ )
5		ecdmn0m 6631	. . . . . 6 |- ( A e. dom .~ <-> E. y y e. [ A ] .~ )
6	5	biimpri 133	. . . . 5 |- ( E. y y e. [ A ] .~ -> A e. dom .~ )
7		ercpbl.r	. . . . . . 7 |- ( ph -> .~ Er V )
8		erdm 6597	. . . . . . 7 |- ( .~ Er V -> dom .~ = V )
9	7, 8	syl 14	. . . . . 6 |- ( ph -> dom .~ = V )
10	9	eleq2d 2263	. . . . 5 |- ( ph -> ( A e. dom .~ <-> A e. V ) )
11	6, 10	imbitrid 154	. . . 4 |- ( ph -> ( E. y y e. [ A ] .~ -> A e. V ) )
12	4, 11	syl5 32	. . 3 |- ( ph -> ( y e. [ A ] .~ -> A e. V ) )
13		eceq1 6622	. . . . . 6 |- ( x = A -> [ x ] .~ = [ A ] .~ )
14		simpr 110	. . . . . 6 |- ( ( ph /\ A e. V ) -> A e. V )
15		ercpbl.v	. . . . . . . 8 |- ( ph -> V e. W )
16	7	ecss 6630	. . . . . . . 8 |- ( ph -> [ A ] .~ C_ V )
17	15, 16	ssexd 4169	. . . . . . 7 |- ( ph -> [ A ] .~ e. _V )
18	17	adantr 276	. . . . . 6 |- ( ( ph /\ A e. V ) -> [ A ] .~ e. _V )
19	1, 13, 14, 18	fvmptd3 5651	. . . . 5 |- ( ( ph /\ A e. V ) -> ( F ` A ) = [ A ] .~ )
20	19	eleq2d 2263	. . . 4 |- ( ( ph /\ A e. V ) -> ( y e. ( F ` A ) <-> y e. [ A ] .~ ) )
21	20	ex 115	. . 3 |- ( ph -> ( A e. V -> ( y e. ( F ` A ) <-> y e. [ A ] .~ ) ) )
22	3, 12, 21	pm5.21ndd 706	. 2 |- ( ph -> ( y e. ( F ` A ) <-> y e. [ A ] .~ ) )
23	22	eqrdv 2191	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   |-> cmpt 4090   dom cdm 4659   ` cfv 5254   Er wer 6584   [cec 6585

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fv 5262  df-er 6587  df-ec 6589

This theorem is referenced by:  qusrhm  14024