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Theorem feq123 5934
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 1053 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐹 = 𝐺)
2 simp2 1054 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
3 simp3 1055 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
41, 2, 3feq123d 5933 1 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030   = wceq 1474  wf 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794
This theorem is referenced by:  feq12i  5937  mbfresfi  32429
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