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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapcod | Structured version Visualization version GIF version |
Description: Compose two mappings. (Contributed by SN, 11-Mar-2025.) |
Ref | Expression |
---|---|
mapcod.1 | ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑m 𝐵)) |
mapcod.2 | ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑m 𝐶)) |
Ref | Expression |
---|---|
mapcod | ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapcod.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑m 𝐵)) | |
2 | elmapex 8867 | . . . 4 ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | simpld 494 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | mapcod.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑m 𝐶)) | |
6 | elmapex 8867 | . . . 4 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
8 | 7 | simprd 495 | . 2 ⊢ (𝜑 → 𝐶 ∈ V) |
9 | elmapi 8868 | . . . 4 ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) → 𝐹:𝐵⟶𝐴) | |
10 | 1, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
11 | elmapi 8868 | . . . 4 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐶) → 𝐺:𝐶⟶𝐵) | |
12 | 5, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵) |
13 | 10, 12 | fcod 6749 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴) |
14 | 4, 8, 13 | elmapdd 8860 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 Vcvv 3471 ∘ ccom 5682 ⟶wf 6544 (class class class)co 7420 ↑m cmap 8845 |
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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8847 |
This theorem is referenced by: evlselv 41820 |
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