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Theorem fcof 6692
Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6693. (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 6501 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 fncofn 6618 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))
32ex 414 . . . . . 6 (𝐹 Fn 𝐴 → (Fun 𝐺 → (𝐹𝐺) Fn (𝐺𝐴)))
43adantr 482 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (Fun 𝐺 → (𝐹𝐺) Fn (𝐺𝐴)))
5 rncoss 5928 . . . . . . 7 ran (𝐹𝐺) ⊆ ran 𝐹
6 sstr 3953 . . . . . . 7 ((ran (𝐹𝐺) ⊆ ran 𝐹 ∧ ran 𝐹𝐵) → ran (𝐹𝐺) ⊆ 𝐵)
75, 6mpan 689 . . . . . 6 (ran 𝐹𝐵 → ran (𝐹𝐺) ⊆ 𝐵)
87adantl 483 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → ran (𝐹𝐺) ⊆ 𝐵)
94, 8jctird 528 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (Fun 𝐺 → ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵)))
109imp 408 . . 3 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵))
111, 10sylanb 582 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵))
12 df-f 6501 . 2 ((𝐹𝐺):(𝐺𝐴)⟶𝐵 ↔ ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵))
1311, 12sylibr 233 1 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → (𝐹𝐺):(𝐺𝐴)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wss 3911  ccnv 5633  ran crn 5635  cima 5637  ccom 5638  Fun wfun 6491   Fn wfn 6492  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  fco  6693  funcofd  6702  f1cof1  6750  focofo  6770  fcores  45387
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