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Theorem foeq1 6024
 Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq1 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))

Proof of Theorem foeq1
StepHypRef Expression
1 fneq1 5893 . . 3 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
2 rneq 5272 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32eqeq1d 2612 . . 3 (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
41, 3anbi12d 743 . 2 (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5 df-fo 5810 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6 df-fo 5810 . 2 (𝐺:𝐴onto𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
74, 5, 63bitr4g 302 1 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ran crn 5039   Fn wfn 5799  –onto→wfo 5802 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-rn 5049  df-fun 5806  df-fn 5807  df-fo 5810 This theorem is referenced by:  f1oeq1  6040  foeq123d  6045  resdif  6070  exfo  6285  fodomr  7996  fowdom  8359  brwdom2  8361  canthp1lem2  9354  mndfo  17138  znzrhfo  19715  pjhfo  27949  elunop  28115  elunop2  28256  nnfoctbdjlem  39348
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