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Theorem foeq2 5454
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 5324 . . 3 (𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
21anbi1d 465 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)))
3 df-fo 5241 . 2 (𝐹:𝐴onto𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶))
4 df-fo 5241 . 2 (𝐹:𝐵onto𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶))
52, 3, 43bitr4g 223 1 (𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  ran crn 4645   Fn wfn 5230  ontowfo 5233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-fn 5238  df-fo 5241
This theorem is referenced by:  f1oeq2  5469  foeq123d  5473  tposfo  6295  ctssdclemr  7140  enumct  7143  exmidfodomrlemr  7230  exmidfodomrlemrALT  7231  ctinf  12480  ctiunct  12490  ssomct  12495  subctctexmid  15204
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