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Theorem funmptd 13420
Description: The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5208, then prove funmptd 13420 from it, and then prove funmpt 5209 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

Hypothesis
Ref Expression
funmptd.def  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
Assertion
Ref Expression
funmptd  |-  ( ph  ->  Fun  F )

Proof of Theorem funmptd
StepHypRef Expression
1 funmpt 5209 . 2  |-  Fun  (
x  e.  A  |->  B )
2 funmptd.def . . 3  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
32funeqd 5193 . 2  |-  ( ph  ->  ( Fun  F  <->  Fun  ( x  e.  A  |->  B ) ) )
41, 3mpbiri 167 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    |-> cmpt 4026   Fun wfun 5165
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-fun 5173
This theorem is referenced by: (None)
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