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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 8838 | . . . 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 8838 | . . . 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 8839 | . . . 4 ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) → 𝐹:𝐵⟶𝐴) | |
10 | 1, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
11 | elmapi 8839 | . . . 4 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐶) → 𝐺:𝐶⟶𝐵) | |
12 | 5, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝐶⟶𝐵) |
13 | 10, 12 | fcod 6733 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐶⟶𝐴) |
14 | 4, 8, 13 | elmapdd 8831 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 Vcvv 3466 ∘ ccom 5670 ⟶wf 6529 (class class class)co 7401 ↑m cmap 8816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-map 8818 |
This theorem is referenced by: evlselv 41648 |
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