Theorem divsfval 12971

Description: Value of the function in qusval 12966. (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 5644	. . . 4 |- ( y e. ( F ` A ) -> A e. V )
3	2	a1i 9	. . 3 |- ( ph -> ( y e. ( F ` A ) -> A e. V ) )
4		19.8a 1604	. . . 4 |- ( y e. [ A ] .~ -> E. y y e. [ A ] .~ )
5		ecdmn0m 6636	. . . . . 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 6602	. . . . . . 7 |- ( .~ Er V -> dom .~ = V )
9	7, 8	syl 14	. . . . . 6 |- ( ph -> dom .~ = V )
10	9	eleq2d 2266	. . . . 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 6627	. . . . . 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 6635	. . . . . . . 8 |- ( ph -> [ A ] .~ C_ V )
17	15, 16	ssexd 4173	. . . . . . 7 |- ( ph -> [ A ] .~ e. _V )
18	17	adantr 276	. . . . . 6 |- ( ( ph /\ A e. V ) -> [ A ] .~ e. _V )
19	1, 13, 14, 18	fvmptd3 5655	. . . . 5 |- ( ( ph /\ A e. V ) -> ( F ` A ) = [ A ] .~ )
20	19	eleq2d 2266	. . . 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 2194	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   |-> cmpt 4094   dom cdm 4663   ` cfv 5258   Er wer 6589   [cec 6590

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fv 5266  df-er 6592  df-ec 6594

This theorem is referenced by:  qusrhm  14084