Theorem curry1f 4105

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 4041	. . . . 5 |- ((F:(A X. B)-->D /\ C e. A /\ x e. B) -> (CFx) e. D)
2	1	3expa 835	. . . 4 |- (((F:(A X. B)-->D /\ C e. A) /\ x e. B) -> (CFx) e. D)
3	2	r19.21aiva 1717	. . 3 |- ((F:(A X. B)-->D /\ C e. A) -> A.x e. B (CFx) e. D)
4		eqid 1478	. . . 4 |- {<.x, y>. | (x e. B /\ y = (CFx))} = {<.x, y>. | (x e. B /\ y = (CFx))}
5	4	fopab2 3829	. . 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 4104	. . . 4 |- ((F Fn (A X. B) /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})
9		ffn 3633	. . . 4 |- (F:(A X. B)-->D -> F Fn (A X. B))
10	8, 9	sylan 450	. . 3 |- ((F:(A X. B)-->D /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})
11	10	feq1d 3630	. 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 958   e. wcel 960  A.wral 1648  Vcvv 1814  {csn 2413  {copab 2671   X. cxp 3174  `'ccnv 3175   |` cres 3178   o. ccom 3180   Fn wfn 3183  -->wf 3184  (class class class)co 3969  2ndc2nd 4084

This theorem is referenced by:  invfval 8257

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-opr 3971  df-2nd 4086

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