Theorem fneq1i 5945

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 5939	. 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 196   = wceq 1480   Fn wfn 5845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-fun 5852  df-fn 5853

This theorem is referenced by:  fnunsn  5958  mptfnf  5974  fnopabg  5976  f1oun  6115  f1oi  6133  f1osn  6135  ovid  6731  curry1  7215  curry2  7218  wfrlem5  7365  wfrlem13  7373  tfrlem10  7429  tfr1  7439  seqomlem2  7492  seqomlem3  7493  seqomlem4  7494  fnseqom  7496  unblem4  8160  r1fnon  8575  alephfnon  8833  alephfplem4  8875  alephfp  8876  cfsmolem  9037  infpssrlem3  9072  compssiso  9141  hsmexlem5  9197  axdclem2  9287  wunex2  9505  wuncval2  9514  om2uzrani  12688  om2uzf1oi  12689  uzrdglem  12693  uzrdgfni  12694  uzrdg0i  12695  hashkf  13056  dmaf  16615  cdaf  16616  prdsinvlem  17440  srg1zr  18445  pws1  18532  frlmphl  20034  ovolunlem1  23167  0plef  23340  0pledm  23341  itg1ge0  23354  itg1addlem4  23367  mbfi1fseqlem5  23387  itg2addlem  23426  qaa  23977  0vfval  27301  xrge0pluscn  29760  bnj927  30539  bnj535  30660  frrlem5  31477  fullfunfnv  31687  neibastop2lem  31989  fourierdlem42  39660  rngcrescrhm  41346  rngcrescrhmALTV  41364