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Theorem fmpt2d 5996
Description: Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
Hypotheses
Ref Expression
fmpt2d.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
fmpt2d.1  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
fmpt2d.3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
Assertion
Ref Expression
fmpt2d  |-  ( ph  ->  F : A --> C )
Distinct variable groups:    x, A    y, A    y, C    y, F    ph, x    ph, y
Allowed substitution hints:    B( x, y)    C( x)    F( x)    V( x, y)

Proof of Theorem fmpt2d
StepHypRef Expression
1 fmpt2d.2 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
21ralrimiva 2817 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  V )
3 eqid 2402 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5646 . . . 4  |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 17 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 fmpt2d.1 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5608 . . 3  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 232 . 2  |-  ( ph  ->  F  Fn  A )
9 fmpt2d.3 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
109ralrimiva 2817 . 2  |-  ( ph  ->  A. y  e.  A  ( F `  y )  e.  C )
11 ffnfv 5992 . 2  |-  ( F : A --> C  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  e.  C
) )
128, 10, 11sylanbrc 662 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753    |-> cmpt 4452    Fn wfn 5520   -->wf 5521   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533
This theorem is referenced by:  cantnff  8045  limsupgre  13360  idaf  15558  curfcl  15717  yonedainv  15766  mat2pmatf  19413  m2cpmf  19427  pm2mpf  19483  clsf  19733  kgenf  20226  rrxcph  22008  lgamf  23589  vmaf  23666  lgsdchr  23896  mirf  24318  omsf  28624  erdszelem6  29374  cdleme50f  33542  dochfN  34357  binomcxplemdvsum  36089
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