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Theorem fcof 6759
Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6760. (Contributed by AV, 18-Sep-2024.)
Assertion
Ref Expression
fcof ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)

Proof of Theorem fcof
StepHypRef Expression
1 df-f 6565 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 fncofn 6685 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))
32ex 412 . . . . . 6 (𝐹 Fn 𝐴 → (Fun 𝐺 → (𝐹𝐺) Fn (𝐺𝐴)))
43adantr 480 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (Fun 𝐺 → (𝐹𝐺) Fn (𝐺𝐴)))
5 rncoss 5986 . . . . . . 7 ran (𝐹𝐺) ⊆ ran 𝐹
6 sstr 3992 . . . . . . 7 ((ran (𝐹𝐺) ⊆ ran 𝐹 ∧ ran 𝐹𝐵) → ran (𝐹𝐺) ⊆ 𝐵)
75, 6mpan 690 . . . . . 6 (ran 𝐹𝐵 → ran (𝐹𝐺) ⊆ 𝐵)
87adantl 481 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → ran (𝐹𝐺) ⊆ 𝐵)
94, 8jctird 526 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (Fun 𝐺 → ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵)))
109imp 406 . . 3 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵))
111, 10sylanb 581 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵))
12 df-f 6565 . 2 ((𝐹𝐺):(𝐺𝐴)⟶𝐵 ↔ ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵))
1311, 12sylibr 234 1 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wss 3951  ccnv 5684  ran crn 5686  cima 5688  ccom 5689  Fun wfun 6555   Fn wfn 6556  wf 6557
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  fco  6760  funcofd  6768  f1cof1  6814  focofo  6833  fcores  47079
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