Theorem curry2val 6895

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 6893	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( x F C ) ) )
3	2	fveq1d 5874	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( ( x e. A |-> ( x F C ) ) ` D ) )
4		eqid 2420	. . . . . . . . . . 11 |- ( x e. A |-> ( x F C ) ) = ( x e. A |-> ( x F C ) )
5	4	dmmptss 5342	. . . . . . . . . 10 |- dom ( x e. A |-> ( x F C ) ) C_ A
6	5	sseli 3457	. . . . . . . . 9 |- ( D e. dom ( x e. A |-> ( x F C ) ) -> D e. A )
7	6	con3i 140	. . . . . . . 8 |- ( -. D e. A -> -. D e. dom ( x e. A |-> ( x F C ) ) )
8		ndmfv 5896	. . . . . . . 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 467	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = (/) )
11		fndm 5684	. . . . . . 7 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
12		simpl 458	. . . . . . . 8 |- ( ( D e. A /\ C e. B ) -> D e. A )
13	12	con3i 140	. . . . . . 7 |- ( -. D e. A -> -. ( D e. A /\ C e. B ) )
14		ndmovg 6457	. . . . . . 7 |- ( ( dom F = ( A X. B ) /\ -. ( D e. A /\ C e. B ) ) -> ( D F C ) = (/) )
15	11, 13, 14	syl2an 479	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( D F C ) = (/) )
16	10, 15	eqtr4d 2464	. . . . 5 |- ( ( F Fn ( A X. B ) /\ -. D e. A ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
17	16	ex 435	. . . 4 |- ( F Fn ( A X. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
18	17	adantr 466	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( -. D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) ) )
19		oveq1 6303	. . . 4 |- ( x = D -> ( x F C ) = ( D F C ) )
20		ovex 6324	. . . 4 |- ( D F C ) e. _V
21	19, 4, 20	fvmpt 5955	. . 3 |- ( D e. A -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
22	18, 21	pm2.61d2 163	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( ( x e. A |-> ( x F C ) ) ` D ) = ( D F C ) )
23	3, 22	eqtrd 2461	1 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( G ` D ) = ( D F C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1867   _Vcvv 3078   (/)c0 3758   {csn 3993   |-> cmpt 4475   X. cxp 4843   `'ccnv 4844   dom cdm 4845   |` cres 4847   o. ccom 4849   Fn wfn 5587   ` cfv 5592  (class class class)co 6296   1stc1st 6796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-1st 6798  df-2nd 6799

This theorem is referenced by:  curry2ima  28126