Theorem fcof 5865

Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5529. (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 5358	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		fncofn 5864	. . . . . . 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 5030	. . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
6		sstr 3248	. . . . . . 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 5358	. 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 3213  ^◡ccnv 4750  ran crn 4752   “ cima 4754   ∘ ccom 4755  Fun wfun 5348   Fn wfn 5349  ⟶wf 5350

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-fun 5356  df-fn 5357  df-f 5358

This theorem is referenced by: (None)