Theorem curry2val 6870

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 6868	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( x F C ) ) )
3	2	fveq1d 5850	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( ( x e. A |-> ( x F C ) ) ` D ) )
4		eqid 2454	. . . . . . . . . . 11 |- ( x e. A |-> ( x F C ) ) = ( x e. A |-> ( x F C ) )
5	4	dmmptss 5486	. . . . . . . . . 10 |- dom ( x e. A |-> ( x F C ) ) C_ A
6	5	sseli 3485	. . . . . . . . 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 5872	. . . . . . . 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 464	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
11		fndm 5662	. . . . . . 7 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
12		simpl 455	. . . . . . . 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 6431	. . . . . . 7 |- ( ( dom F = ( A X. B ) /\ -. ( D e. A /\ C e. B ) ) -> ( D F C ) = (/) )
15	11, 13, 14	syl2an 475	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( D F C ) = (/) )
16	10, 15	eqtr4d 2498	. . . . 5 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
17	16	ex 432	. . . 4 |- ( F Fn ( A X. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
18	17	adantr 463	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
19		oveq1 6277	. . . 4 |- ( x = D -> ( x F C ) = ( D F C ) )
20		ovex 6298	. . . 4 |- ( D F C ) e. _V
21	19, 4, 20	fvmpt 5931	. . 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 2495	1 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( D F C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   (/)c0 3783   {csn 4016   |-> cmpt 4497   X. cxp 4986   `'ccnv 4987   dom cdm 4988   |` cres 4990   o. ccom 4992   Fn wfn 5565   ` cfv 5570  (class class class)co 6270   1stc1st 6771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-1st 6773  df-2nd 6774

This theorem is referenced by:  curry2ima  27755