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Theorem feq123 5395
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 999 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐹 = 𝐺)
2 simp2 1000 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
3 simp3 1001 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
41, 2, 3feq123d 5394 1 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  w3a 980   = wceq 1364  wf 5250
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258
This theorem is referenced by: (None)
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