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Mirrors > Home > MPE Home > Th. List > joinfval2 | Structured version Visualization version GIF version |
Description: Value of join function for a poset-type structure. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) |
Ref | Expression |
---|---|
joinfval.u | ⊢ 𝑈 = (lub‘𝐾) |
joinfval.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
joinfval2 | ⊢ (𝐾 ∈ 𝑉 → ∨ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinfval.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
2 | joinfval.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | 1, 2 | joinfval 17599 | . 2 ⊢ (𝐾 ∈ 𝑉 → ∨ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦}𝑈𝑧}) |
4 | 1 | lubfun 17578 | . . . . 5 ⊢ Fun 𝑈 |
5 | funbrfv2b 6716 | . . . . 5 ⊢ (Fun 𝑈 → ({𝑥, 𝑦}𝑈𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧))) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ({𝑥, 𝑦}𝑈𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧)) |
7 | eqcom 2825 | . . . . 5 ⊢ ((𝑈‘{𝑥, 𝑦}) = 𝑧 ↔ 𝑧 = (𝑈‘{𝑥, 𝑦})) | |
8 | 7 | anbi2i 622 | . . . 4 ⊢ (({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧) ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))) |
9 | 6, 8 | bitri 276 | . . 3 ⊢ ({𝑥, 𝑦}𝑈𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))) |
10 | 9 | oprabbii 7210 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦}𝑈𝑧} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} |
11 | 3, 10 | syl6eq 2869 | 1 ⊢ (𝐾 ∈ 𝑉 → ∨ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cpr 4559 class class class wbr 5057 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 {coprab 7146 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: joindm 17601 joinval 17603 |
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