Theorem fneq1i 5899

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 5893	. 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 195   = wceq 1475   Fn wfn 5799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807

This theorem is referenced by:  fnunsn  5912  mptfnf  5928  fnopabg  5930  f1oun  6069  f1oi  6086  f1osn  6088  ovid  6675  curry1  7156  curry2  7159  wfrlem5  7306  wfrlem13  7314  tfrlem10  7370  tfr1  7380  seqomlem2  7433  seqomlem3  7434  seqomlem4  7435  fnseqom  7437  unblem4  8100  r1fnon  8513  alephfnon  8771  alephfplem4  8813  alephfp  8814  cfsmolem  8975  infpssrlem3  9010  compssiso  9079  hsmexlem5  9135  axdclem2  9225  wunex2  9439  wuncval2  9448  om2uzrani  12613  om2uzf1oi  12614  uzrdglem  12618  uzrdgfni  12619  uzrdg0i  12620  hashkf  12981  dmaf  16522  cdaf  16523  prdsinvlem  17347  srg1zr  18352  pws1  18439  frlmphl  19939  ovolunlem1  23072  0plef  23245  0pledm  23246  itg1ge0  23259  itg1addlem4  23272  mbfi1fseqlem5  23292  itg2addlem  23331  qaa  23882  2trllemD  26087  eupap1  26503  0vfval  26845  xrge0pluscn  29314  bnj927  30093  bnj535  30214  frrlem5  31028  fullfunfnv  31223  neibastop2lem  31525  fourierdlem42  39042  rngcrescrhm  41877  rngcrescrhmALTV  41896