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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 6546 | . . 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 6537 | . 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 5534 dom cdm 5535 ran crn 5536 “ cima 5538 ∘ ccom 5539 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 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-fun 6351 df-fn 6352 df-f 6353 |
This theorem is referenced by: smfco 43915 |
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