Theorem fcof 6731

Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6732. (Contributed by AV, 18-Sep-2024.)

Assertion

Ref	Expression
fcof	⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)

Proof of Theorem fcof

Step	Hyp	Ref	Expression
1		df-f 6538	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		fncofn 6657	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
3	2	ex 412	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (Fun 𝐺 → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)))
4	3	adantr 480	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (Fun 𝐺 → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)))
5		rncoss 5962	. . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
6		sstr 3983	. . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ∘ 𝐺) ⊆ 𝐵)
7	5, 6	mpan 687	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 → ran (𝐹 ∘ 𝐺) ⊆ 𝐵)
8	7	adantl 481	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ∘ 𝐺) ⊆ 𝐵)
9	4, 8	jctird 526	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (Fun 𝐺 → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵)))
10	9	imp 406	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
11	1, 10	sylanb 580	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
12		df-f 6538	. 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ↔ ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
13	11, 12	sylibr 233	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3941  ^◡ccnv 5666  ran crn 5668   “ cima 5670   ∘ ccom 5671  Fun wfun 6528   Fn wfn 6529  ⟶wf 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-fun 6536  df-fn 6537  df-f 6538

This theorem is referenced by:  fco  6732  funcofd  6741  f1cof1  6789  focofo  6809  fcores  46287