Theorem curry1val 4038

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

Hypothesis

Ref	Expression
curry1.1	|- G = (F o. `'(2nd |` ({C} X. V)))

Assertion

Ref	Expression
curry1val	|- ((F Fn (A X. B) /\ C e. A /\ D e. U) -> (G` D) = (CFD))

Proof of Theorem curry1val

Step	Hyp	Ref	Expression
1		curry1.1	. . . . 5 |- G = (F o. `'(2nd |` ({C} X. V)))
2	1	curry1 4036	. . . 4 |- ((F Fn (A X. B) /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})
3	2	fveq1d 3665	. . 3 |- ((F Fn (A X. B) /\ C e. A) -> (G` D) = ({<.x, y>. | (x e. B /\ y = (CFx))}` D))
4	3	3adant3 796	. 2 |- ((F Fn (A X. B) /\ C e. A /\ D e. U) -> (G` D) = ({<.x, y>. | (x e. B /\ y = (CFx))}` D))
5		eqid 1452	. . . . . . . 8 |- {<.x, y>. | (x e. B /\ y = (CFx))} = {<.x, y>. | (x e. B /\ y = (CFx))}
6	5	fvopab4ndm 3723	. . . . . . 7 |- (-. D e. B -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (/))
7	6	3ad2ant3 799	. . . . . 6 |- ((F Fn (A X. B) /\ D e. U /\ -. D e. B) -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (/))
8		ndmoprg 3982	. . . . . . 7 |- ((dom F = (A X. B) /\ D e. U /\ -. (C e. A /\ D e. B)) -> (CFD) = (/))
9		fndm 3527	. . . . . . 7 |- (F Fn (A X. B) -> dom F = (A X. B))
10		id 59	. . . . . . 7 |- (D e. U -> D e. U)
11		pm3.27 323	. . . . . . . 8 |- ((C e. A /\ D e. B) -> D e. B)
12	11	con3i 98	. . . . . . 7 |- (-. D e. B -> -. (C e. A /\ D e. B))
13	8, 9, 10, 12	syl3an 865	. . . . . 6 |- ((F Fn (A X. B) /\ D e. U /\ -. D e. B) -> (CFD) = (/))
14	7, 13	eqtr4d 1486	. . . . 5 |- ((F Fn (A X. B) /\ D e. U /\ -. D e. B) -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (CFD))
15	14	3expia 832	. . . 4 |- ((F Fn (A X. B) /\ D e. U) -> (-. D e. B -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (CFD)))
16		opreq2 3908	. . . . 5 |- (x = D -> (CFx) = (CFD))
17		oprex 3922	. . . . 5 |- (CFD) e. V
18	16, 5, 17	fvopab4 3719	. . . 4 |- (D e. B -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (CFD))
19	15, 18	pm2.61d2 129	. . 3 |- ((F Fn (A X. B) /\ D e. U) -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (CFD))
20	19	3adant2 795	. 2 |- ((F Fn (A X. B) /\ C e. A /\ D e. U) -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (CFD))
21	4, 20	eqtrd 1483	1 |- ((F Fn (A X. B) /\ C e. A /\ D e. U) -> (G` D) = (CFD))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 772   = wceq 1099   e. wcel 1105  Vcvv 1786  (/)c0 2251  {csn 2380  {copab 2634   X. cxp 3131  `'ccnv 3132  dom cdm 3133   |` cres 3135   o. ccom 3137   Fn wfn 3140  ` cfv 3145  (class class class)co 3902  2ndc2nd 4016

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-opr 3904  df-2nd 4018

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