Theorem fneq1i 6445

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 6439	. 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 208   = wceq 1533   Fn wfn 6345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-fun 6352  df-fn 6353

This theorem is referenced by:  fnunsn  6459  mptfnf  6478  fnopabg  6480  f1oun  6629  f1oi  6647  f1osn  6649  ovid  7285  curry1  7793  curry2  7796  fsplitfpar  7808  wfrlem5  7953  wfrlem13  7961  tfrlem10  8017  tfr1  8027  seqomlem2  8081  seqomlem3  8082  seqomlem4  8083  fnseqom  8085  unblem4  8767  r1fnon  9190  alephfnon  9485  alephfplem4  9527  alephfp  9528  cfsmolem  9686  infpssrlem3  9721  compssiso  9790  hsmexlem5  9846  axdclem2  9936  wunex2  10154  wuncval2  10163  om2uzrani  13314  om2uzf1oi  13315  uzrdglem  13319  uzrdgfni  13320  uzrdg0i  13321  hashkf  13686  dmaf  17303  cdaf  17304  prdsinvlem  18202  srg1zr  19273  pws1  19360  frlmphl  20919  ovolunlem1  24092  0plef  24267  0pledm  24268  itg1ge0  24281  itg1addlem4  24294  mbfi1fseqlem5  24314  itg2addlem  24353  qaa  24906  ex-fpar  28235  0vfval  28377  xrge0pluscn  31178  bnj927  32035  bnj535  32157  frrlem11  33128  fullfunfnv  33402  neibastop2lem  33703  fourierdlem42  42427  rngcrescrhm  44349  rngcrescrhmALTV  44367