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Theorem fnmptd 13421
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 2530 . . 3  |-  ( ph  ->  A. x  e.  A  B  e.  V )
3 eqid 2157 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5297 . . 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 5261 . 2  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 166 1  |-  ( ph  ->  F  Fn  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   A.wral 2435    |-> cmpt 4026    Fn wfn 5166
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-fun 5173  df-fn 5174
This theorem is referenced by: (None)
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