Theorem divsfval 13625

Description: Value of the function in qusval 13620. (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 5765	. . . 4 |- ( y e. ( F ` A ) -> A e. V )
3	2	a1i 9	. . 3 |- ( ph -> ( y e. ( F ` A ) -> A e. V ) )
4		19.8a 1639	. . . 4 |- ( y e. [ A ] .~ -> E. y y e. [ A ] .~ )
5		ecdmn0m 6824	. . . . . 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 6790	. . . . . . 7 |- ( .~ Er V -> dom .~ = V )
9	7, 8	syl 14	. . . . . 6 |- ( ph -> dom .~ = V )
10	9	eleq2d 2304	. . . . 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 6815	. . . . . 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 6823	. . . . . . . 8 |- ( ph -> [ A ] .~ C_ V )
17	15, 16	ssexd 4255	. . . . . . 7 |- ( ph -> [ A ] .~ e. _V )
18	17	adantr 276	. . . . . 6 |- ( ( ph /\ A e. V ) -> [ A ] .~ e. _V )
19	1, 13, 14, 18	fvmptd3 5776	. . . . 5 |- ( ( ph /\ A e. V ) -> ( F ` A ) = [ A ] .~ )
20	19	eleq2d 2304	. . . 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 713	. 2 |- ( ph -> ( y e. ( F ` A ) <-> y e. [ A ] .~ ) )
23	22	eqrdv 2232	1 |- ( ph -> ( F ` A ) = [ A ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   |-> cmpt 4176   dom cdm 4754   ` cfv 5357   Er wer 6777   [cec 6778

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fv 5365  df-er 6780  df-ec 6782

This theorem is referenced by:  qusrhm  14788