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Theorem fcof 6740
Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6741. (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 6546 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 fncofn 6665 . . . . . . 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 5969 . . . . . . 7 ran (𝐹𝐺) ⊆ ran 𝐹
6 sstr 3986 . . . . . . 7 ((ran (𝐹𝐺) ⊆ ran 𝐹 ∧ ran 𝐹𝐵) → ran (𝐹𝐺) ⊆ 𝐵)
75, 6mpan 689 . . . . . 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 580 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ∧ ran (𝐹𝐺) ⊆ 𝐵))
12 df-f 6546 . 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 395  wss 3945  ccnv 5671  ran crn 5673  cima 5675  ccom 5676  Fun wfun 6536   Fn wfn 6537  wf 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6544  df-fn 6545  df-f 6546
This theorem is referenced by:  fco  6741  funcofd  6750  f1cof1  6798  focofo  6818  fcores  46443
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