MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmpt2d Unicode version

Theorem fmpt2d 5830
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 2725 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  V )
3 eqid 2380 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5504 . . . 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 5469 . . 3  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 224 . 2  |-  ( ph  ->  F  Fn  A )
9 fmpt2d.3 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )
109ralrimiva 2725 . 2  |-  ( ph  ->  A. y  e.  A  ( F `  y )  e.  C )
11 ffnfv 5826 . 2  |-  ( F : A --> C  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  e.  C
) )
128, 10, 11sylanbrc 646 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642    e. cmpt 4200    Fn wfn 5382   -->wf 5383   ` cfv 5387
This theorem is referenced by:  fmpt2dOLD  5831  cantnff  7555  limsupgre  12195  idaf  14138  curfcl  14249  yonedainv  14298  clsf  17028  kgenf  17487  vmaf  20762  lgsdchr  20992  lgamf  24598  erdszelem6  24654  cdleme50f  30707  dochfN  31522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395
  Copyright terms: Public domain W3C validator