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Theorem foeq2 5342
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 5212 . . 3 (𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
21anbi1d 460 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)))
3 df-fo 5129 . 2 (𝐹:𝐴onto𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶))
4 df-fo 5129 . 2 (𝐹:𝐵onto𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶))
52, 3, 43bitr4g 222 1 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  ran crn 4540   Fn wfn 5118  ontowfo 5121
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-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-fn 5126  df-fo 5129
This theorem is referenced by:  f1oeq2  5357  foeq123d  5361  tposfo  6168  ctssdclemr  6997  enumct  7000  exmidfodomrlemr  7063  exmidfodomrlemrALT  7064  ctinf  11954  ctiunct  11964  subctctexmid  13255
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