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Theorem foeq1 5130
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 5015 . . 3 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
2 rneq 4589 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32eqeq1d 2064 . . 3 (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
41, 3anbi12d 450 . 2 (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5 df-fo 4936 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6 df-fo 4936 . 2 (𝐺:𝐴onto𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
74, 5, 63bitr4g 216 1 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  ran crn 4374   Fn wfn 4925  ontowfo 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-fo 4936
This theorem is referenced by:  f1oeq1  5145  foeq123d  5150  resdif  5176  dif1en  6368
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