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Theorem foeq2 6799
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq2 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))

Proof of Theorem foeq2
StepHypRef Expression
1 fneq2 6638 . . 3 (𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
21anbi1d 631 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)))
3 df-fo 6546 . 2 (𝐹:𝐴onto𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶))
4 df-fo 6546 . 2 (𝐹:𝐵onto𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶))
52, 3, 43bitr4g 314 1 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  ran crn 5676   Fn wfn 6535  ontowfo 6538
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-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-fn 6543  df-fo 6546
This theorem is referenced by:  foco  6816  f1oeq2  6819  foeq123d  6823  tposfo  8233  brwdom  9558  brwdom2  9564  canthwdom  9570  cfslb2n  10259  0ramcl  16952  ghmcyg  19756  txcmpb  23130  qtoptopon  23190  fsupprnfi  31892  opidon2OLD  36660  fnfocofob  45722  fullthinc  47568
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