Theorem fopab2ga 10455

Description: Functionality and domain of a class given by the "maps to" notation.

Hypothesis

Ref	Expression
fopab2ga.1	|- F = (x e. A |-> C)

Assertion

Ref	Expression
fopab2ga	|- (A.x e. A C e. V <-> F Fn A)

Distinct variable group:   x,A

Proof of Theorem fopab2ga

Step	Hyp	Ref	Expression
1		fopab2ga.1	. . 3 |- F = (x e. A |-> C)
2		df-mpt 4079	. . 3 |- (x e. A |-> C) = {<.x, y>. | (x e. A /\ y = C)}
3	1, 2	eqtr 1498	. 2 |- F = {<.x, y>. | (x e. A /\ y = C)}
4	3	fnopab2g 3622	1 |- (A.x e. A C e. V <-> F Fn A)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814  {copab 2671   Fn wfn 3183   e. cmpt 4077

This theorem is referenced by:  cmpdom 10458  cnvtr 10609

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-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-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-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-fun 3198  df-fn 3199  df-mpt 4079

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