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Theorem feq123 6485
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 1133 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐹 = 𝐺)
2 simp2 1134 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
3 simp3 1135 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
41, 2, 3feq123d 6484 1 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wf 6332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-fun 6338  df-fn 6339  df-f 6340
This theorem is referenced by:  feq12i  6488  hashfxnn0  13691  mbfresfi  35003
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