Theorem fcof 6693

Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6694. (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 6503	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		fncofn 6617	. . . . . . 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 5928	. . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
6		sstr 3952	. . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ∘ 𝐺) ⊆ 𝐵)
7	5, 6	mpan 690	. . . . . 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 581	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
12		df-f 6503	. 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ↔ ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))
13	11, 12	sylibr 234	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3911  ^◡ccnv 5630  ran crn 5632   “ cima 5634   ∘ ccom 5635  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6501  df-fn 6502  df-f 6503

This theorem is referenced by:  fco  6694  funcofd  6702  f1cof1  6748  focofo  6767  fcores  47061