Theorem fcof 5833

Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5500. (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 5330	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		fncofn 5832	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
3	2	ex 115	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (Fun 𝐺 → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)))
4	3	adantr 276	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (Fun 𝐺 → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)))
5		rncoss 5003	. . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
6		sstr 3235	. . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ∘ 𝐺) ⊆ 𝐵)
7	5, 6	mpan 424	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 → ran (𝐹 ∘ 𝐺) ⊆ 𝐵)
8	7	adantl 277	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ∘ 𝐺) ⊆ 𝐵)
9	4, 8	jctird 317	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (Fun 𝐺 → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵)))
10	9	imp 124	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
11	1, 10	sylanb 284	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
12		df-f 5330	. 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ↔ ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
13	11, 12	sylibr 134	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ⊆ wss 3200  ^◡ccnv 4724  ran crn 4726   “ cima 4728   ∘ ccom 4729  Fun wfun 5320   Fn wfn 5321  ⟶wf 5322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by: (None)