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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 1203 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → 𝑅 ∈ 𝑊) | |
3 | simpl 484 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) → 𝑋 ∈ 𝑉) | |
4 | simpl 484 | . . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) → 𝑌 ∈ 𝑉) | |
5 | simpl 484 | . . . 4 ⊢ ((𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊) → 𝑋 ≠ 𝑌) | |
6 | 3, 4, 5 | 3anim123i 1152 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ≠ 𝑌)) |
7 | simpr 486 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
8 | simpr 486 | . . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) → 𝐵 ∈ 𝑅) | |
9 | 7, 8 | anim12i 614 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅)) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅)) |
10 | 9 | 3adant3 1133 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅)) |
11 | fprmappr 46511 | . . 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 2838 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 {cpr 4592 ⟨cop 4596 (class class class)co 7361 ↑m cmap 8771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 |
This theorem is referenced by: lincvalpr 46589 ldepspr 46644 |
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