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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 17017 | . . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺}) |
5 | 4 | eleq2d 2687 | . . 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 4268 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦}) | |
10 | 9 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺)) |
11 | preq2 4269 | . . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌}) | |
12 | 11 | eleq1d 2686 | . . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
13 | 10, 12 | opelopabg 4993 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
14 | 7, 8, 13 | syl2anc 693 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
15 | 6, 14 | bitrd 268 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 {cpr 4179 〈cop 4183 {copab 4712 dom cdm 5114 ‘cfv 5888 glbcglb 16943 meetcmee 16945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-oprab 6654 df-glb 16975 df-meet 16977 |
This theorem is referenced by: meetval 17019 meetcl 17020 meetdmss 17021 meeteu 17024 clatl 17116 |
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