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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 6566 . . 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 5988 . . . . . . 7 ran (𝐹𝐺) ⊆ ran 𝐹
6 sstr 4003 . . . . . . 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 6566 . 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 3962  ccnv 5687  ran crn 5689  cima 5691  ccom 5692  Fun wfun 6556   Fn wfn 6557  wf 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564  df-fn 6565  df-f 6566
This theorem is referenced by:  fco  6760  funcofd  6768  f1cof1  6814  focofo  6833  fcores  47016
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