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Theorem foeq3 5435
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 2187 . . 3 (𝐴 = 𝐵 → (ran 𝐹 = 𝐴 ↔ ran 𝐹 = 𝐵))
21anbi2d 464 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵)))
3 df-fo 5221 . 2 (𝐹:𝐶onto𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴))
4 df-fo 5221 . 2 (𝐹:𝐶onto𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵))
52, 3, 43bitr4g 223 1 (𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  ran crn 4626   Fn wfn 5210  ontowfo 5213
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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-fo 5221
This theorem is referenced by:  f1oeq3  5450  foeq123d  5453  resdif  5482  ffoss  5492  fifo  6976  enumct  7111  ctssexmid  7145  exmidfodomrlemr  7198  exmidfodomrlemrALT  7199  qnnen  12424  ctiunctal  12434  unct  12435  subctctexmid  14610
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