MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foeq2 Structured version   Visualization version   GIF version

Theorem foeq2 6754
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 6595 . . 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 6503 . 2 (𝐹:𝐴onto𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶))
4 df-fo 6503 . 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 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-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2729  df-fn 6500  df-fo 6503
This theorem is referenced by:  foco  6771  f1oeq2  6774  foeq123d  6778  tposfo  8185  brwdom  9504  brwdom2  9510  canthwdom  9516  cfslb2n  10205  0ramcl  16896  ghmcyg  19674  txcmpb  22998  qtoptopon  23058  fsupprnfi  31610  opidon2OLD  36316  fnfocofob  45318  fullthinc  47073
  Copyright terms: Public domain W3C validator