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Theorem fnmpt2 5770
Description: Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
Hypothesis
Ref Expression
fmpt2.1  F  ,  |->  C
Assertion
Ref Expression
fnmpt2  C  V  F  Fn  X.
Distinct variable groups:   ,,   ,,
Allowed substitution hints:    C(,)    F(,)    V(,)

Proof of Theorem fnmpt2
StepHypRef Expression
1 elex 2560 . . . 4  C  V  C  _V
21ralimi 2378 . . 3  C  V  C  _V
32ralimi 2378 . 2  C  V  C  _V
4 fmpt2.1 . . . 4  F  ,  |->  C
54fmpt2 5769 . . 3  C  _V  F :  X.  --> _V
6 dffn2 4990 . . 3  F  Fn  X.  F :  X. 
--> _V
75, 6bitr4i 176 . 2  C  _V  F  Fn  X.
83, 7sylib 127 1  C  V  F  Fn  X.
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551    X. cxp 4286    Fn wfn 4840   -->wf 4841    |-> cmpt2 5457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  fnmpt2i  5772  divfnzn  8332  cnref1o  8357
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