Theorem curry2val 5080

Description: The value of a curried function with a constant second argument.

Hypothesis

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

Assertion

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

Proof of Theorem curry2val

Step	Hyp	Ref	Expression
1		curry2.1	. . . . 5 |- G = (F o. `'(1st |` (_V X. {C})))
2	1	curry2 5078	. . . 4 |- ((F Fn (A X. B) /\ C e. B) -> G = {<.x, y>. | (x e. A /\ y = (xFC))})
3	2	fveq1d 4683	. . 3 |- ((F Fn (A X. B) /\ C e. B) -> (G` D) = ({<.x, y>. | (x e. A /\ y = (xFC))}` D))
4	3	3adant3 896	. 2 |- ((F Fn (A X. B) /\ C e. B /\ D e. U) -> (G` D) = ({<.x, y>. | (x e. A /\ y = (xFC))}` D))
5		eqid 1884	. . . . . . . 8 |- {<.x, y>. | (x e. A /\ y = (xFC))} = {<.x, y>. | (x e. A /\ y = (xFC))}
6	5	fvopab4ndm 4747	. . . . . . 7 |- (-. D e. A -> ({<.x, y>. | (x e. A /\ y = (xFC))}` D) = (/))
7	6	3ad2ant3 899	. . . . . 6 |- ((F Fn (A X. B) /\ C e. B /\ -. D e. A) -> ({<.x, y>. | (x e. A /\ y = (xFC))}` D) = (/))
8		ndmoprg 4976	. . . . . . 7 |- ((dom F = (A X. B) /\ C e. B /\ -. (D e. A /\ C e. B)) -> (DFC) = (/))
9		fndm 4512	. . . . . . 7 |- (F Fn (A X. B) -> dom F = (A X. B))
10		id 73	. . . . . . 7 |- (C e. B -> C e. B)
11		simpl 346	. . . . . . . 8 |- ((D e. A /\ C e. B) -> D e. A)
12	11	con3i 114	. . . . . . 7 |- (-. D e. A -> -. (D e. A /\ C e. B))
13	8, 9, 10, 12	syl3an 1139	. . . . . 6 |- ((F Fn (A X. B) /\ C e. B /\ -. D e. A) -> (DFC) = (/))
14	7, 13	eqtr4d 1928	. . . . 5 |- ((F Fn (A X. B) /\ C e. B /\ -. D e. A) -> ({<.x, y>. | (x e. A /\ y = (xFC))}` D) = (DFC))
15	14	3expia 1069	. . . 4 |- ((F Fn (A X. B) /\ C e. B) -> (-. D e. A -> ({<.x, y>. | (x e. A /\ y = (xFC))}` D) = (DFC)))
16		opreq1 4889	. . . . 5 |- (x = D -> (xFC) = (DFC))
17		oprex 4907	. . . . 5 |- (DFC) e. _V
18	16, 5, 17	fvopab4 4743	. . . 4 |- (D e. A -> ({<.x, y>. | (x e. A /\ y = (xFC))}` D) = (DFC))
19	15, 18	pm2.61d2 143	. . 3 |- ((F Fn (A X. B) /\ C e. B) -> ({<.x, y>. | (x e. A /\ y = (xFC))}` D) = (DFC))
20	19	3adant3 896	. 2 |- ((F Fn (A X. B) /\ C e. B /\ D e. U) -> ({<.x, y>. | (x e. A /\ y = (xFC))}` D) = (DFC))
21	4, 20	eqtrd 1925	1 |- ((F Fn (A X. B) /\ C e. B /\ D e. U) -> (G` D) = (DFC))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  {csn 3044  {copab 3395   X. cxp 3984  `'ccnv 3985  dom cdm 3986   |` cres 3988   o. ccom 3990   Fn wfn 3993  ` cfv 3998  (class class class)co 4884  1stc1st 5018

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021

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