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Theorem fncofn 6667
Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 6668. (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 6650 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 funco 6589 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
31, 2sylan 581 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → Fun (𝐹𝐺))
43funfnd 6580 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn dom (𝐹𝐺))
5 fndm 6653 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 482 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → dom 𝐹 = 𝐴)
76eqcomd 2739 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → 𝐴 = dom 𝐹)
87imaeq2d 6060 . . . 4 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = (𝐺 “ dom 𝐹))
9 dmco 6254 . . . 4 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
108, 9eqtr4di 2791 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = dom (𝐹𝐺))
1110fneq2d 6644 . 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 5676  dom cdm 5677  cima 5680  ccom 5681  Fun wfun 6538   Fn wfn 6539
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547
This theorem is referenced by:  fnco  6668  fcof  6741
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