|   | Mathbox for Steven Nguyen | < Previous  
      Next > Nearby theorems | |
| 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 8889 | . . . 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 8889 | . . . 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 8890 | . . . 4 ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) → 𝐹:𝐵⟶𝐴) | |
| 10 | 1, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) | 
| 11 | elmapi 8890 | . . . 4 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐶) → 𝐺:𝐶⟶𝐵) | |
| 12 | 5, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵) | 
| 13 | 10, 12 | fcod 6760 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴) | 
| 14 | 4, 8, 13 | elmapdd 8882 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 ∘ ccom 5688 ⟶wf 6556 (class class class)co 7432 ↑m cmap 8867 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 | 
| This theorem is referenced by: evlselv 42602 | 
| Copyright terms: Public domain | W3C validator |