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Theorem foeq1 5348
 Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq1 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))

Proof of Theorem foeq1
StepHypRef Expression
1 fneq1 5218 . . 3 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
2 rneq 4773 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32eqeq1d 2149 . . 3 (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
41, 3anbi12d 465 . 2 (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5 df-fo 5136 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6 df-fo 5136 . 2 (𝐺:𝐴onto𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
74, 5, 63bitr4g 222 1 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ran crn 4547   Fn wfn 5125  –onto→wfo 5128 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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-fun 5132  df-fn 5133  df-fo 5136 This theorem is referenced by:  f1oeq1  5363  foeq123d  5368  resdif  5396  dif1en  6780  0ct  6999  ctmlemr  7000  ctm  7001  ctssdclemn0  7002  ctssdclemr  7004  ctssdc  7005  enumct  7007  omct  7009  ctssexmid  7031  exmidfodomrlemim  7073  ennnfonelemim  11971  ctinfomlemom  11974  ctinfom  11975  ctinf  11977  qnnen  11978  enctlem  11979  ctiunct  11987  omctfn  11990  subctctexmid  13367
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