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Theorem foeq3 5387
 Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq3 (𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))

Proof of Theorem foeq3
StepHypRef Expression
1 eqeq2 2167 . . 3 (𝐴 = 𝐵 → (ran 𝐹 = 𝐴 ↔ ran 𝐹 = 𝐵))
21anbi2d 460 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵)))
3 df-fo 5173 . 2 (𝐹:𝐶onto𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴))
4 df-fo 5173 . 2 (𝐹:𝐶onto𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵))
52, 3, 43bitr4g 222 1 (𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335  ran crn 4584   Fn wfn 5162  –onto→wfo 5165 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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-fo 5173 This theorem is referenced by:  f1oeq3  5402  foeq123d  5405  resdif  5433  ffoss  5443  fifo  6917  enumct  7049  ctssexmid  7076  exmidfodomrlemr  7120  exmidfodomrlemrALT  7121  qnnen  12132  ctiunctal  12142  unct  12143  subctctexmid  13534
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