Theorem curry1val 4093

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 4091	. . . 4 |- ((F Fn (A X. B) /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})
3	2	fveq1d 3721	. . 3 |- ((F Fn (A X. B) /\ C e. A) -> (G` D) = ({<.x, y>. | (x e. B /\ y = (CFx))}` D))
4	3	3adant3 798	. 2 |- ((F Fn (A X. B) /\ C e. A /\ D e. U) -> (G` D) = ({<.x, y>. | (x e. B /\ y = (CFx))}` D))
5		eqid 1474	. . . . . . . 8 |- {<.x, y>. | (x e. B /\ y = (CFx))} = {<.x, y>. | (x e. B /\ y = (CFx))}
6	5	fvopab4ndm 3779	. . . . . . 7 |- (-. D e. B -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (/))
7	6	3ad2ant3 801	. . . . . 6 |- ((F Fn (A X. B) /\ D e. U /\ -. D e. B) -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (/))
8		ndmoprg 4038	. . . . . . 7 |- ((dom F = (A X. B) /\ D e. U /\ -. (C e. A /\ D e. B)) -> (CFD) = (/))
9		fndm 3583	. . . . . . 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 867	. . . . . 6 |- ((F Fn (A X. B) /\ D e. U /\ -. D e. B) -> (CFD) = (/))
14	7, 13	eqtr4d 1508	. . . . 5 |- ((F Fn (A X. B) /\ D e. U /\ -. D e. B) -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (CFD))
15	14	3expia 834	. . . 4 |- ((F Fn (A X. B) /\ D e. U) -> (-. D e. B -> ({<.x, y>. | (x e. B /\ y = (CFx))}` D) = (CFD)))
16		opreq2 3964	. . . . 5 |- (x = D -> (CFx) = (CFD))
17		oprex 3978	. . . . 5 |- (CFD) e. V
18	16, 5, 17	fvopab4 3775	. . . 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 797	. 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 1505	1 |- ((F Fn (A X. B) /\ C e. A /\ D e. U) -> (G` D) = (CFD))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  Vcvv 1808  (/)c0 2277  {csn 2406  {copab 2662   X. cxp 3164  `'ccnv 3165  dom cdm 3166   |` cres 3168   o. ccom 3170   Fn wfn 3173  ` cfv 3178  (class class class)co 3958  2ndc2nd 4071

This theorem is referenced by:  invfval 8225  hhssabl 9087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-opr 3960  df-2nd 4073

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