Theorem cmpdom 10468

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 3193	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
2		cmpdom.1	. . . 4 |- F = (x e. A |-> B)
3	2	fopab2ga 10465	. . 3 |- (A.x e. A B e. V <-> F Fn A)
4		ancom 435	. . 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 10467	. 2 |- Fun F
7	5, 6	mpbiran2 729	1 |- (A.x e. A B e. V <-> dom F = A)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  dom cdm 3170  Fun wfun 3176   Fn wfn 3177   e. cmpt 4071

This theorem is referenced by:  trdom 10635

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-mpt 4073

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