Theorem dmmptg 4972

Description: The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)

Assertion

Ref	Expression
dmmptg	|- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   V( x)

Proof of Theorem dmmptg

Step	Hyp	Ref	Expression
1		elex 2652	. . . 4 |- ( B e. V -> B e. _V )
2	1	ralimi 2454	. . 3 |- ( A. x e. A B e. V -> A. x e. A B e. _V )
3		rabid2 2565	. . 3 |- ( A = { x e. A | B e. _V } <-> A. x e. A B e. _V )
4	2, 3	sylibr 133	. 2 |- ( A. x e. A B e. V -> A = { x e. A | B e. _V } )
5		eqid 2100	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
6	5	dmmpt 4970	. 2 |- dom ( x e. A |-> B ) = { x e. A | B e. _V }
7	4, 6	syl6reqr 2151	1 |- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )

Syntax hints:   -> wi 4   = wceq 1299   e. wcel 1448   A.wral 2375   {crab 2379   _Vcvv 2641   |-> cmpt 3929   dom cdm 4477

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-mpt 3931  df-xp 4483  df-rel 4484  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490

This theorem is referenced by:  resfunexg  5573  rdgtfr  6201  rdgruledefgg  6202  negfi  10838