ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fneq1 GIF version

Theorem fneq1 5258
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
fneq1 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Proof of Theorem fneq1
StepHypRef Expression
1 funeq 5190 . . 3 (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺))
2 dmeq 4786 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
32eqeq1d 2166 . . 3 (𝐹 = 𝐺 → (dom 𝐹 = 𝐴 ↔ dom 𝐺 = 𝐴))
41, 3anbi12d 465 . 2 (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴)))
5 df-fn 5173 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6 df-fn 5173 . 2 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
74, 5, 63bitr4g 222 1 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  dom cdm 4586  Fun wfun 5164   Fn wfn 5165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-fun 5172  df-fn 5173
This theorem is referenced by:  fneq1d  5260  fneq1i  5264  fn0  5289  feq1  5302  foeq1  5388  f1ocnv  5427  mpteqb  5558  eufnfv  5697  tfr0dm  6269  tfrlemiex  6278  tfr1onlemsucfn  6287  tfr1onlemsucaccv  6288  tfr1onlembxssdm  6290  tfr1onlembfn  6291  tfr1onlemex  6294  tfr1onlemaccex  6295  tfr1onlemres  6296  mapval2  6623  elixp2  6647  ixpfn  6649  elixpsn  6680  cc2lem  7186  cc3  7188
  Copyright terms: Public domain W3C validator