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Theorem feq123 6269
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 1172 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐹 = 𝐺)
2 simp2 1173 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
3 simp3 1174 . 2 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
41, 2, 3feq123d 6268 1 ((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1113   = wceq 1658  wf 6120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-fun 6126  df-fn 6127  df-f 6128
This theorem is referenced by:  feq12i  6272  hashfxnn0  13418  mbfresfi  34000
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