Theorem dmmpo 6434

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 6433	. 2 |- F Fn ( A X. B )
4		fndm 5478	. 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 1402   e. wcel 2209   _Vcvv 2821   X. cxp 4770   dom cdm 4772   Fn wfn 5370   e. cmpo 6081

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369

This theorem is referenced by:  genipdm  7877