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Theorem fnmptf 26121
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypothesis
Ref Expression
mptfnf.0  |-  F/_ x A
Assertion
Ref Expression
fnmptf  |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )

Proof of Theorem fnmptf
StepHypRef Expression
1 elex 3080 . . 3  |-  ( B  e.  V  ->  B  e.  _V )
21ralimi 2814 . 2  |-  ( A. x  e.  A  B  e.  V  ->  A. x  e.  A  B  e.  _V )
3 mptfnf.0 . . 3  |-  F/_ x A
43mptfnf 26120 . 2  |-  ( A. x  e.  A  B  e.  _V  <->  ( x  e.  A  |->  B )  Fn  A )
52, 4sylib 196 1  |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   F/_wnfc 2599   A.wral 2795   _Vcvv 3071    |-> cmpt 4451    Fn wfn 5514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-fun 5521  df-fn 5522
This theorem is referenced by:  offval2f  26127  esumc  26643
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