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Theorem fncofn 6671
Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 6672. (Contributed by AV, 17-Sep-2024.)
Assertion
Ref Expression
fncofn ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))

Proof of Theorem fncofn
StepHypRef Expression
1 fnfun 6654 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 funco 6593 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
31, 2sylan 579 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → Fun (𝐹𝐺))
43funfnd 6584 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn dom (𝐹𝐺))
5 fndm 6657 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 480 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → dom 𝐹 = 𝐴)
76eqcomd 2734 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → 𝐴 = dom 𝐹)
87imaeq2d 6063 . . . 4 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = (𝐺 “ dom 𝐹))
9 dmco 6258 . . . 4 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
108, 9eqtr4di 2786 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = dom (𝐹𝐺))
1110fneq2d 6648 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ↔ (𝐹𝐺) Fn dom (𝐹𝐺)))
124, 11mpbird 257 1 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  ccnv 5677  dom cdm 5678  cima 5681  ccom 5682  Fun wfun 6542   Fn wfn 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-fun 6550  df-fn 6551
This theorem is referenced by:  fnco  6672  fcof  6746
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