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Theorem foeq1 6753
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 6594 . . 3 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
2 rneq 5892 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32eqeq1d 2739 . . 3 (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
41, 3anbi12d 632 . 2 (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5 df-fo 6503 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6 df-fo 6503 . 2 (𝐺:𝐴onto𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
74, 5, 63bitr4g 314 1 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  ran crn 5635   Fn wfn 6492  ontowfo 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-fo 6503
This theorem is referenced by:  fimadmfoALT  6768  f1oeq1  6773  foeq123d  6778  resdif  6806  exfo  7056  mapfoss  8791  fodomr  9073  dif1enlem  9101  dif1enlemOLD  9102  fowdom  9508  brwdom2  9510  canthp1lem2  10590  mndfo  18581  sursubmefmnd  18707  znzrhfo  20957  pjhfo  30651  elunop  30817  elunop2  30958  symgcom  31937  nnfoctbdjlem  44703  fcoreslem3  45306  fcoresfo  45312  fcoresfob  45313  fundcmpsurbijinjpreimafv  45606
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