MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnmpt2 Unicode version

Theorem fnmpt2 6192
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  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
fnmpt2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B
) )
Distinct variable groups:    x, A, y    x, B, y
Allowed substitution hints:    C( x, y)    F( x, y)    V( x, y)

Proof of Theorem fnmpt2
StepHypRef Expression
1 elex 2796 . . . 4  |-  ( C  e.  V  ->  C  e.  _V )
21ralimi 2618 . . 3  |-  ( A. y  e.  B  C  e.  V  ->  A. y  e.  B  C  e.  _V )
32ralimi 2618 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  A. x  e.  A  A. y  e.  B  C  e.  _V )
4 fmpt2.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
54fmpt2 6191 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  e.  _V  <->  F : ( A  X.  B ) --> _V )
6 dffn2 5390 . . 3  |-  ( F  Fn  ( A  X.  B )  <->  F :
( A  X.  B
) --> _V )
75, 6bitr4i 243 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  _V  <->  F  Fn  ( A  X.  B ) )
83, 7sylib 188 1  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    X. cxp 4687    Fn wfn 5250   -->wf 5251    e. cmpt2 5860
This theorem is referenced by:  fnmpt2i  6193  genpdm  8626  1stfcl  13971  2ndfcl  13972  cnre2csqima  23295  mpt2cti  23346  dya2iocseg  23579  cbcpcp  25162  dmmpt2g  27981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
  Copyright terms: Public domain W3C validator