Theorem cmpdom 10458

Description: Domain of a class given by the "maps to" notation.

Hypothesis

Ref	Expression
cmpdom.1	|- F = (x e. A |-> B)

Assertion

Ref	Expression
cmpdom	|- (A.x e. A B e. V <-> dom F = A)

Distinct variable group:   x,A

Proof of Theorem cmpdom

Step	Hyp	Ref	Expression
1		df-fn 3199	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
2		cmpdom.1	. . . 4 |- F = (x e. A |-> B)
3	2	fopab2ga 10455	. . 3 |- (A.x e. A B e. V <-> F Fn A)
4		ancom 437	. . 3 |- ((dom F = A /\ Fun F) <-> (Fun F /\ dom F = A))
5	1, 3, 4	3bitr4 183	. 2 |- (A.x e. A B e. V <-> (dom F = A /\ Fun F))
6	2	cmpfun 10457	. 2 |- Fun F
7	5, 6	mpbiran2 731	1 |- (A.x e. A B e. V <-> dom F = A)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814  dom cdm 3176  Fun wfun 3182   Fn wfn 3183   e. cmpt 4077

This theorem is referenced by:  trdom 10606

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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