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Theorem feq123 5065
Description: Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
Assertion
Ref Expression
feq123 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))

Proof of Theorem feq123
StepHypRef Expression
1 simp1 915 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐹 = 𝐺)
2 simp2 916 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
3 simp3 917 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
41, 2, 3feq123d 5064 1 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  w3a 896   = wceq 1259  wf 4925
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 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-fun 4931  df-fn 4932  df-f 4933
This theorem is referenced by: (None)
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