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Mirrors > Home > MPE Home > Th. List > fcof | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fcof | ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 6546 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | fncofn 6665 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)) | |
3 | 2 | ex 412 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (Fun 𝐺 → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))) |
4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (Fun 𝐺 → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))) |
5 | rncoss 5969 | . . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 | |
6 | sstr 3986 | . . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ∘ 𝐺) ⊆ 𝐵) | |
7 | 5, 6 | mpan 689 | . . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 → ran (𝐹 ∘ 𝐺) ⊆ 𝐵) |
8 | 7 | adantl 481 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ∘ 𝐺) ⊆ 𝐵) |
9 | 4, 8 | jctird 526 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (Fun 𝐺 → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵))) |
10 | 9 | imp 406 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵)) |
11 | 1, 10 | sylanb 580 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵)) |
12 | df-f 6546 | . 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ↔ ((𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴) ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐵)) | |
13 | 11, 12 | sylibr 233 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶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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