Theorem curry2val 6890

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 6888	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( x F C ) ) )
3	2	fveq1d 5873	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( ( x e. A |-> ( x F C ) ) ` D ) )
4		eqid 2467	. . . . . . . . . . 11 |- ( x e. A |-> ( x F C ) ) = ( x e. A |-> ( x F C ) )
5	4	dmmptss 5508	. . . . . . . . . 10 |- dom ( x e. A |-> ( x F C ) ) C_ A
6	5	sseli 3505	. . . . . . . . 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 5895	. . . . . . . 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 5685	. . . . . . 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 6452	. . . . . . 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 2511	. . . . 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 6301	. . . 4 |- ( x = D -> ( x F C ) = ( D F C ) )
20		ovex 6319	. . . 4 |- ( D F C ) e. _V
21	19, 4, 20	fvmpt 5956	. . 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 2508	1 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( D F C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3118   (/)c0 3790   {csn 4032   |-> cmpt 4510   X. cxp 5002   `'ccnv 5003   dom cdm 5004   |` cres 5006   o. ccom 5008   Fn wfn 5588   ` cfv 5593  (class class class)co 6294   1stc1st 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-1st 6794  df-2nd 6795

This theorem is referenced by:  curry2ima  27317