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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 16840 | . . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺}) |
5 | 4 | eleq2d 2673 | . . 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 4212 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦}) | |
10 | 9 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺)) |
11 | preq2 4213 | . . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌}) | |
12 | 11 | eleq1d 2672 | . . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
13 | 10, 12 | opelopabg 4918 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
14 | 7, 8, 13 | syl2anc 691 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
15 | 6, 14 | bitrd 267 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cpr 4127 〈cop 4131 {copab 4642 dom cdm 5038 ‘cfv 5804 glbcglb 16766 meetcmee 16768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-oprab 6553 df-glb 16798 df-meet 16800 |
This theorem is referenced by: meetval 16842 meetcl 16843 meetdmss 16844 meeteu 16847 clatl 16939 |
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