Theorem elmapex 6647

Description: Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
elmapex	|- ( A e. ( B ^m C ) -> ( B e. _V /\ C e. _V ) )

Proof of Theorem elmapex

Dummy variables x f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-map 6628	. 2 |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
2	1	elmpocl 6047	1 |- ( A e. ( B ^m C ) -> ( B e. _V /\ C e. _V ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   {cab 2156   _Vcvv 2730   -->wf 5194  (class class class)co 5853   ^m cmap 6626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628

This theorem is referenced by:  elmapi  6648  elmapssres  6651  mapsspm  6660  mapss  6669