ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqfnfvd Unicode version

Theorem eqfnfvd 5416
Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
eqfnfvd.1  |-  ( ph  ->  F  Fn  A )
eqfnfvd.2  |-  ( ph  ->  G  Fn  A )
eqfnfvd.3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
Assertion
Ref Expression
eqfnfvd  |-  ( ph  ->  F  =  G )
Distinct variable groups:    x, A    x, F    x, G    ph, x

Proof of Theorem eqfnfvd
StepHypRef Expression
1 eqfnfvd.3 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
21ralrimiva 2447 . 2  |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
3 eqfnfvd.1 . . 3  |-  ( ph  ->  F  Fn  A )
4 eqfnfvd.2 . . 3  |-  ( ph  ->  G  Fn  A )
5 eqfnfv 5413 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
63, 4, 5syl2anc 404 . 2  |-  ( ph  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
72, 6mpbird 166 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   A.wral 2360    Fn wfn 5025   ` cfv 5030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-csb 2937  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fn 5033  df-fv 5038
This theorem is referenced by:  foeqcnvco  5585  f1eqcocnv  5586  tfrlem1  6089  frecrdg  6189  updjudhcoinlf  6827  updjudhcoinrg  6828  iseqvalt  9936  seq3val  9937  iseqoveq  9948  iseqsst  9949  iseqfeq2  9954  seq3feq2  9956  iseqfeq  9959  seq3shft  10335  efcvgfsum  11020  peano4nninf  12199  nninfalllemn  12201  nninfsellemeqinf  12211
  Copyright terms: Public domain W3C validator