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Mirrors > Home > MPE Home > Th. List > meetdef | Structured version Visualization version GIF version |
Description: Two ways to say that a meet is defined. (Contributed by NM, 9-Sep-2018.) |
Ref | Expression |
---|---|
meetdef.u | ⊢ 𝐺 = (glb‘𝐾) |
meetdef.m | ⊢ ∧ = (meet‘𝐾) |
meetdef.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
meetdef.x | ⊢ (𝜑 → 𝑋 ∈ 𝑊) |
meetdef.y | ⊢ (𝜑 → 𝑌 ∈ 𝑍) |
Ref | Expression |
---|---|
meetdef | ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meetdef.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
2 | meetdef.u | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
3 | meetdef.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
4 | 2, 3 | meetdm 18446 | . . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺}) |
5 | 4 | eleq2d 2824 | . . 3 ⊢ (𝐾 ∈ 𝑉 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺})) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺})) |
7 | meetdef.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊) | |
8 | meetdef.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑍) | |
9 | preq1 4737 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦}) | |
10 | 9 | eleq1d 2823 | . . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺)) |
11 | preq2 4738 | . . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌}) | |
12 | 11 | eleq1d 2823 | . . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
13 | 10, 12 | opelopabg 5547 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
14 | 7, 8, 13 | syl2anc 584 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
15 | 6, 14 | bitrd 279 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 {cpr 4632 〈cop 4636 {copab 5209 dom cdm 5688 ‘cfv 6562 glbcglb 18367 meetcmee 18369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-oprab 7434 df-glb 18404 df-meet 18406 |
This theorem is referenced by: meetval 18448 meetcl 18449 meetdmss 18450 meeteu 18453 clatl 18565 meetdm2 48766 |
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