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Theorem fncofn 6654
Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 6655. (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 6639 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 funco 6578 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
31, 2sylan 580 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → Fun (𝐹𝐺))
43funfnd 6569 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn dom (𝐹𝐺))
5 fndm 6642 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 481 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → dom 𝐹 = 𝐴)
76eqcomd 2738 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → 𝐴 = dom 𝐹)
87imaeq2d 6050 . . . 4 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = (𝐺 “ dom 𝐹))
9 dmco 6243 . . . 4 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
108, 9eqtr4di 2790 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = dom (𝐹𝐺))
1110fneq2d 6633 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ↔ (𝐹𝐺) Fn dom (𝐹𝐺)))
124, 11mpbird 256 1 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  ccnv 5669  dom cdm 5670  cima 5673  ccom 5674  Fun wfun 6527   Fn wfn 6528
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-fun 6535  df-fn 6536
This theorem is referenced by:  fnco  6655  fcof  6728
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