Theorem curry2val 6893

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 6891	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( x F C ) ) )
3	2	fveq1d 5867	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( ( x e. A |-> ( x F C ) ) ` D ) )
4		eqid 2451	. . . . . . . . . . 11 |- ( x e. A |-> ( x F C ) ) = ( x e. A |-> ( x F C ) )
5	4	dmmptss 5331	. . . . . . . . . 10 |- dom ( x e. A |-> ( x F C ) ) C_ A
6	5	sseli 3428	. . . . . . . . 9 |- ( D e. dom ( x e. A |-> ( x F C ) ) -> D e. A )
7	6	con3i 141	. . . . . . . 8 |- ( -. D e. A -> -. D e. dom ( x e. A |-> ( x F C ) ) )
8		ndmfv 5889	. . . . . . . 8 |- ( -. D e. dom ( x e. A |-> ( x F C ) ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
9	7, 8	syl 17	. . . . . . 7 |- ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
10	9	adantl 468	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
11		fndm 5675	. . . . . . 7 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
12		simpl 459	. . . . . . . 8 |- ( ( D e. A /\ C e. B ) -> D e. A )
13	12	con3i 141	. . . . . . 7 |- ( -. D e. A -> -. ( D e. A /\ C e. B ) )
14		ndmovg 6452	. . . . . . 7 |- ( ( dom F = ( A X. B ) /\ -. ( D e. A /\ C e. B ) ) -> ( D F C ) = (/) )
15	11, 13, 14	syl2an 480	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( D F C ) = (/) )
16	10, 15	eqtr4d 2488	. . . . 5 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
17	16	ex 436	. . . 4 |- ( F Fn ( A X. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
18	17	adantr 467	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
19		oveq1 6297	. . . 4 |- ( x = D -> ( x F C ) = ( D F C ) )
20		ovex 6318	. . . 4 |- ( D F C ) e. _V
21	19, 4, 20	fvmpt 5948	. . 3 |- ( D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
22	18, 21	pm2.61d2 164	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
23	3, 22	eqtrd 2485	1 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( D F C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   (/)c0 3731   {csn 3968   |-> cmpt 4461   X. cxp 4832   `'ccnv 4833   dom cdm 4834   |` cres 4836   o. ccom 4838   Fn wfn 5577   ` cfv 5582  (class class class)co 6290   1stc1st 6791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-1st 6793  df-2nd 6794

This theorem is referenced by:  curry2ima  28289