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Mirrors > Home > MPE Home > Th. List > funcofd | Structured version Visualization version GIF version |
Description: Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.) |
Ref | Expression |
---|---|
funcofd.1 | ⊢ (𝜑 → Fun 𝐹) |
funcofd.2 | ⊢ (𝜑 → Fun 𝐺) |
Ref | Expression |
---|---|
funcofd | ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcofd.1 | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
2 | fdmrn 6555 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) | |
3 | 1, 2 | sylib 221 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹) |
4 | funcofd.2 | . 2 ⊢ (𝜑 → Fun 𝐺) | |
5 | fcof 6546 | . 2 ⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) | |
6 | 3, 4, 5 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ◡ccnv 5535 dom cdm 5536 ran crn 5537 “ cima 5539 ∘ ccom 5540 Fun wfun 6352 ⟶wf 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: smfco 43951 |
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