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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 6397 | . . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2anc 586 | . . . 4 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
5 | fdmrn 6540 | . . . 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 6109 | . . . 4 ⊢ dom (𝐹 ∘ 𝐺) = (◡𝐺 “ dom 𝐹) | |
8 | 7 | feq2i 6508 | . . 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 5845 | . . 3 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹) |
12 | 9, 11 | fssd 6530 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3938 ◡ccnv 5556 dom cdm 5557 ran crn 5558 “ cima 5560 ∘ ccom 5561 Fun wfun 6351 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: smfco 43084 |
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