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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mapprop | Structured version Visualization version GIF version | ||
| Description: An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.) (Proof shortened by AV, 2-Jun-2024.) | 
| Ref | Expression | 
|---|---|
| mapprop.f | ⊢ 𝐹 = {〈𝑋, 𝐴〉, 〈𝑌, 𝐵〉} | 
| Ref | Expression | 
|---|---|
| mapprop | ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → 𝐹 ∈ (𝑅 ↑m {𝑋, 𝑌})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mapprop.f | . 2 ⊢ 𝐹 = {〈𝑋, 𝐴〉, 〈𝑌, 𝐵〉} | |
| 2 | simp3r 1202 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → 𝑅 ∈ 𝑊) | |
| 3 | simpl 482 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) → 𝑋 ∈ 𝑉) | |
| 4 | simpl 482 | . . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) → 𝑌 ∈ 𝑉) | |
| 5 | simpl 482 | . . . 4 ⊢ ((𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊) → 𝑋 ≠ 𝑌) | |
| 6 | 3, 4, 5 | 3anim123i 1151 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ≠ 𝑌)) | 
| 7 | simpr 484 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
| 8 | simpr 484 | . . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) → 𝐵 ∈ 𝑅) | |
| 9 | 7, 8 | anim12i 613 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅)) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅)) | 
| 10 | 9 | 3adant3 1132 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅)) | 
| 11 | fprmappr 48266 | . . 3 ⊢ ((𝑅 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ≠ 𝑌) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅)) → {〈𝑋, 𝐴〉, 〈𝑌, 𝐵〉} ∈ (𝑅 ↑m {𝑋, 𝑌})) | |
| 12 | 2, 6, 10, 11 | syl3anc 1372 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → {〈𝑋, 𝐴〉, 〈𝑌, 𝐵〉} ∈ (𝑅 ↑m {𝑋, 𝑌})) | 
| 13 | 1, 12 | eqeltrid 2844 | 1 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → 𝐹 ∈ (𝑅 ↑m {𝑋, 𝑌})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 {cpr 4627 〈cop 4631 (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-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 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-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 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-map 8869 | 
| This theorem is referenced by: lincvalpr 48340 ldepspr 48395 | 
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