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Theorem fmpt2d 5985
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 2830 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  V )
3 eqid 2454 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5648 . . . 4  |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 16 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 fmpt2d.1 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5612 . . 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 2830 . 2  |-  ( ph  ->  A. y  e.  A  ( F `  y )  e.  C )
11 ffnfv 5981 . 2  |-  ( F : A --> C  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  e.  C
) )
128, 10, 11sylanbrc 664 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    |-> cmpt 4461    Fn wfn 5524   -->wf 5525   ` cfv 5529
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537
This theorem is referenced by:  cantnff  7997  limsupgre  13081  idaf  15054  curfcl  15165  yonedainv  15214  clsf  18794  kgenf  19256  rrxcph  21038  vmaf  22600  lgsdchr  22830  mirf  23215  sgnsf  26364  lgamf  27195  erdszelem6  27251  mat2pmatf  31240  m2cpmf  31254  pm2mpf  31308  cdleme50f  34549  dochfN  35364
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