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Theorem fnmptd 14438
Description: The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
Hypotheses
Ref Expression
fnmptd.def  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
fnmptd.ex  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
Assertion
Ref Expression
fnmptd  |-  ( ph  ->  F  Fn  A )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    F( x)    V( x)

Proof of Theorem fnmptd
StepHypRef Expression
1 fnmptd.ex . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
21ralrimiva 2550 . . 3  |-  ( ph  ->  A. x  e.  A  B  e.  V )
3 eqid 2177 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5342 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 14 . 2  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 fnmptd.def . . 3  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5306 . 2  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 167 1  |-  ( ph  ->  F  Fn  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455    |-> cmpt 4064    Fn wfn 5211
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-fun 5218  df-fn 5219
This theorem is referenced by: (None)
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