Theorem curry1f 4037

Description: Functionality of a curried function with a constant first argument.

Hypothesis

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

Assertion

Ref	Expression
curry1f	|- ((F:(A X. B)-->D /\ C e. A) -> G:B-->D)

Proof of Theorem curry1f

Step	Hyp	Ref	Expression
1		foprrn 3974	. . . . 5 |- ((F:(A X. B)-->D /\ C e. A /\ x e. B) -> (CFx) e. D)
2	1	3expa 830	. . . 4 |- (((F:(A X. B)-->D /\ C e. A) /\ x e. B) -> (CFx) e. D)
3	2	r19.21aiva 1690	. . 3 |- ((F:(A X. B)-->D /\ C e. A) -> A.x e. B (CFx) e. D)
4		eqid 1452	. . . 4 |- {<.x, y>. | (x e. B /\ y = (CFx))} = {<.x, y>. | (x e. B /\ y = (CFx))}
5	4	fopab2 3762	. . 3 |- (A.x e. B (CFx) e. D <-> {<.x, y>. | (x e. B /\ y = (CFx))}:B-->D)
6	3, 5	sylib 198	. 2 |- ((F:(A X. B)-->D /\ C e. A) -> {<.x, y>. | (x e. B /\ y = (CFx))}:B-->D)
7		curry1.1	. . . . 5 |- G = (F o. `'(2nd |` ({C} X. V)))
8	7	curry1 4036	. . . 4 |- ((F Fn (A X. B) /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})
9		ffn 3567	. . . 4 |- (F:(A X. B)-->D -> F Fn (A X. B))
10	8, 9	sylan 448	. . 3 |- ((F:(A X. B)-->D /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})
11	10	feq1d 3564	. 2 |- ((F:(A X. B)-->D /\ C e. A) -> (G:B-->D <-> {<.x, y>. | (x e. B /\ y = (CFx))}:B-->D))
12	6, 11	mpbird 196	1 |- ((F:(A X. B)-->D /\ C e. A) -> G:B-->D)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 1099   e. wcel 1105  A.wral 1621  Vcvv 1786  {csn 2380  {copab 2634   X. cxp 3131  `'ccnv 3132   |` cres 3135   o. ccom 3137   Fn wfn 3140  -->wf 3141  (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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