Theorem curry2val 6878

Description: The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Hypothesis

Ref	Expression
curry2.1	|- G = ( F o. `' ( 1st |` ( _V X. { C } ) ) )

Assertion

Ref	Expression
curry2val	|- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( D F C ) )

Proof of Theorem curry2val

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curry2.1	. . . 4 |- G = ( F o. `' ( 1st |` ( _V X. { C } ) ) )
2	1	curry2 6876	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( x F C ) ) )
3	2	fveq1d 5854	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( ( x e. A |-> ( x F C ) ) ` D ) )
4		eqid 2441	. . . . . . . . . . 11 |- ( x e. A |-> ( x F C ) ) = ( x e. A |-> ( x F C ) )
5	4	dmmptss 5489	. . . . . . . . . 10 |- dom ( x e. A |-> ( x F C ) ) C_ A
6	5	sseli 3482	. . . . . . . . 9 |- ( D e. dom ( x e. A |-> ( x F C ) ) -> D e. A )
7	6	con3i 135	. . . . . . . 8 |- ( -. D e. A -> -. D e. dom ( x e. A |-> ( x F C ) ) )
8		ndmfv 5876	. . . . . . . 8 |- ( -. D e. dom ( x e. A |-> ( x F C ) ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
9	7, 8	syl 16	. . . . . . 7 |- ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
10	9	adantl 466	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
11		fndm 5666	. . . . . . 7 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
12		simpl 457	. . . . . . . 8 |- ( ( D e. A /\ C e. B ) -> D e. A )
13	12	con3i 135	. . . . . . 7 |- ( -. D e. A -> -. ( D e. A /\ C e. B ) )
14		ndmovg 6439	. . . . . . 7 |- ( ( dom F = ( A X. B ) /\ -. ( D e. A /\ C e. B ) ) -> ( D F C ) = (/) )
15	11, 13, 14	syl2an 477	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( D F C ) = (/) )
16	10, 15	eqtr4d 2485	. . . . 5 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
17	16	ex 434	. . . 4 |- ( F Fn ( A X. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
18	17	adantr 465	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
19		oveq1 6284	. . . 4 |- ( x = D -> ( x F C ) = ( D F C ) )
20		ovex 6305	. . . 4 |- ( D F C ) e. _V
21	19, 4, 20	fvmpt 5937	. . 3 |- ( D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
22	18, 21	pm2.61d2 160	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
23	3, 22	eqtrd 2482	1 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( D F C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1381   e. wcel 1802   _Vcvv 3093   (/)c0 3767   {csn 4010   |-> cmpt 4491   X. cxp 4983   `'ccnv 4984   dom cdm 4985   |` cres 4987   o. ccom 4989   Fn wfn 5569   ` cfv 5574  (class class class)co 6277   1stc1st 6779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-1st 6781  df-2nd 6782

This theorem is referenced by:  curry2ima  27391