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Mirrors > Home > MPE Home > Th. List > joindef | Structured version Visualization version GIF version |
Description: Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018.) |
Ref | Expression |
---|---|
joindef.u | ⊢ 𝑈 = (lub‘𝐾) |
joindef.j | ⊢ ∨ = (join‘𝐾) |
joindef.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
joindef.x | ⊢ (𝜑 → 𝑋 ∈ 𝑊) |
joindef.y | ⊢ (𝜑 → 𝑌 ∈ 𝑍) |
Ref | Expression |
---|---|
joindef | ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joindef.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
2 | joindef.u | . . . . 5 ⊢ 𝑈 = (lub‘𝐾) | |
3 | joindef.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
4 | 2, 3 | joindm 17601 | . . . 4 ⊢ (𝐾 ∈ 𝑉 → dom ∨ = {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈}) |
5 | 4 | eleq2d 2895 | . . 3 ⊢ (𝐾 ∈ 𝑉 → (〈𝑋, 𝑌〉 ∈ dom ∨ ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈})) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈})) |
7 | joindef.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊) | |
8 | joindef.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑍) | |
9 | preq1 4661 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦}) | |
10 | 9 | eleq1d 2894 | . . . 4 ⊢ (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑦} ∈ dom 𝑈)) |
11 | preq2 4662 | . . . . 5 ⊢ (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌}) | |
12 | 11 | eleq1d 2894 | . . . 4 ⊢ (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈)) |
13 | 10, 12 | opelopabg 5416 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑍) → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈)) |
14 | 7, 8, 13 | syl2anc 584 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈)) |
15 | 6, 14 | bitrd 280 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {cpr 4559 〈cop 4563 {copab 5119 dom cdm 5548 ‘cfv 6348 lubclub 17540 joincjn 17542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-oprab 7149 df-lub 17572 df-join 17574 |
This theorem is referenced by: joinval 17603 joincl 17604 joindmss 17605 joineu 17608 clatl 17714 |
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