Theorem dmmpo 6402

Description: Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)

Hypotheses

Ref	Expression
fmpo.1	|- F = ( x e. A , y e. B |-> C )
fnmpoi.2	|- C e. _V

Assertion

Ref	Expression
dmmpo	|- dom F = ( A X. B )

Distinct variable groups:   x, A, y   x, B, y

Allowed substitution hints:   C( x, y)   F( x, y)

Proof of Theorem dmmpo

Step	Hyp	Ref	Expression
1		fmpo.1	. . 3 |- F = ( x e. A , y e. B |-> C )
2		fnmpoi.2	. . 3 |- C e. _V
3	1, 2	fnmpoi 6401	. 2 |- F Fn ( A X. B )
4		fndm 5457	. 2 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
5	3, 4	ax-mp 5	1 |- dom F = ( A X. B )

Syntax hints:   = wceq 1398   e. wcel 2205   _Vcvv 2815   X. cxp 4749   dom cdm 4751   Fn wfn 5349   e. cmpo 6054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337

This theorem is referenced by:  genipdm  7833