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Mirrors > Home > MPE Home > Th. List > Mathboxes > fco3 | Structured version Visualization version GIF version |
Description: Functionality of a composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
fco3.1 | ⊢ (𝜑 → Fun 𝐹) |
fco3.2 | ⊢ (𝜑 → Fun 𝐺) |
Ref | Expression |
---|---|
fco3 | ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fco3.1 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
2 | fco3.2 | . . . . 5 ⊢ (𝜑 → Fun 𝐺) | |
3 | funco 6389 | . . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2anc 586 | . . . 4 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
5 | fdmrn 6532 | . . . 4 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺)) | |
6 | 4, 5 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺)) |
7 | dmco 6101 | . . . 4 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹) | |
8 | 7 | feq2i 6500 | . . 3 ⊢ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶ran (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺)) |
9 | 6, 8 | sylib 220 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran (𝐹 ∘ 𝐺)) |
10 | rncoss 5837 | . . 3 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹) |
12 | 9, 11 | fssd 6522 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3935 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ∘ ccom 5553 Fun wfun 6343 ⟶wf 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 df-fn 6352 df-f 6353 |
This theorem is referenced by: smfco 43071 |
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