Theorem fneq1i 6622

Description: Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
fneq1i.1	⊢ 𝐹 = 𝐺

Assertion

Ref	Expression
fneq1i	⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)

Proof of Theorem fneq1i

Step	Hyp	Ref	Expression
1		fneq1i.1	. 2 ⊢ 𝐹 = 𝐺
2		fneq1 6616	. 2 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)

Syntax hints:   ↔ wb 209   = wceq 1563   Fn wfn 6520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-fun 6527  df-fn 6528

This theorem is referenced by:  fnunop  6641  mptfnf  6660  fnopabg  6662  f1oun  6830  f1oiOLD  6850  f1osn  6852  ovid  7541  curry1  8087  curry2  8090  fsplitfpar  8101  frrlem11  8281  tfrlem10  8362  tfr1  8372  seqomlem2  8426  seqomlem3  8427  seqomlem4  8428  fnseqom  8430  unblem4  9243  r1fnon  9727  alephfnon  10037  alephfplem4  10079  alephfp  10080  cfsmolem  10242  infpssrlem3  10277  compssiso  10346  hsmexlem5  10402  axdclem2  10492  wunex2  10711  wuncval2  10720  om2uzrani  13976  om2uzf1oi  13977  uzrdglem  13981  uzrdgfni  13982  uzrdg0i  13983  hashkf  14356  dmaf  18094  cdaf  18095  prdsinvlem  19103  rng1zrlem  20247  pws1  20394  rngcrescrhm  20757  frlmphl  21888  ovolunlem1  25613  0plef  25788  0pledm  25789  itg1ge0  25802  mbfi1fseqlem5  25835  itg2addlem  25874  qaa  26441  precsexlem1  28354  precsexlem2  28355  precsexlem3  28356  precsexlem4  28357  precsexlem5  28358  ex-fpar  30718  0vfval  30863  xrge0pluscn  34242  bnj927  35070  bnj535  35190  fullfunfnv  36304  neibastop2lem  36728  fnmptif  45839  fourierdlem42  46722  cjnpoly  47482  fcoreslem4  47659  upgrimwlklem1  48518  rngcrescrhmALTV  48901  isofval2  49662