Theorem fncofn 6548

Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 6549. (Contributed by AV, 17-Sep-2024.)

Assertion

Ref	Expression
fncofn	⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))

Proof of Theorem fncofn

Step	Hyp	Ref	Expression
1		fnfun 6533	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		funco 6474	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
3	1, 2	sylan 580	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	3	funfnd 6465	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn dom (𝐹 ∘ 𝐺))
5		fndm 6536	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	adantr 481	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → dom 𝐹 = 𝐴)
7	6	eqcomd 2744	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → 𝐴 = dom 𝐹)
8	7	imaeq2d 5969	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (◡𝐺 “ 𝐴) = (◡𝐺 “ dom 𝐹))
9		dmco 6158	. . . 4 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹)
10	8, 9	eqtr4di 2796	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (◡𝐺 “ 𝐴) = dom (𝐹 ∘ 𝐺))
11	10	fneq2d 6527	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ↔ (𝐹 ∘ 𝐺) Fn dom (𝐹 ∘ 𝐺)))
12	4, 11	mpbird 256	1 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ^◡ccnv 5588  dom cdm 5589   “ cima 5592   ∘ ccom 5593  Fun wfun 6427   Fn wfn 6428

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435  df-fn 6436

This theorem is referenced by:  fnco  6549  fcof  6623