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Theorem fncofn 6618
Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 6619. (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 6603 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 funco 6542 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
31, 2sylan 581 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → Fun (𝐹𝐺))
43funfnd 6533 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn dom (𝐹𝐺))
5 fndm 6606 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 482 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → dom 𝐹 = 𝐴)
76eqcomd 2743 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → 𝐴 = dom 𝐹)
87imaeq2d 6014 . . . 4 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = (𝐺 “ dom 𝐹))
9 dmco 6207 . . . 4 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
108, 9eqtr4di 2795 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = dom (𝐹𝐺))
1110fneq2d 6597 . 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 397   = wceq 1542  ccnv 5633  dom cdm 5634  cima 5637  ccom 5638  Fun wfun 6491   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500
This theorem is referenced by:  fnco  6619  fcof  6692
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