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Mirrors > Home > MPE Home > Th. List > meetfval2 | Structured version Visualization version GIF version |
Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) |
Ref | Expression |
---|---|
meetfval.u | ⊢ 𝐺 = (glb‘𝐾) |
meetfval.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
meetfval2 | ⊢ (𝐾 ∈ 𝑉 → ∧ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meetfval.u | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
2 | meetfval.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
3 | 1, 2 | meetfval 17625 | . 2 ⊢ (𝐾 ∈ 𝑉 → ∧ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦}𝐺𝑧}) |
4 | 1 | glbfun 17603 | . . . . 5 ⊢ Fun 𝐺 |
5 | funbrfv2b 6723 | . . . . 5 ⊢ (Fun 𝐺 → ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧))) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧)) |
7 | eqcom 2828 | . . . . 5 ⊢ ((𝐺‘{𝑥, 𝑦}) = 𝑧 ↔ 𝑧 = (𝐺‘{𝑥, 𝑦})) | |
8 | 7 | anbi2i 624 | . . . 4 ⊢ (({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧) ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))) |
9 | 6, 8 | bitri 277 | . . 3 ⊢ ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))) |
10 | 9 | oprabbii 7221 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦}𝐺𝑧} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))} |
11 | 3, 10 | syl6eq 2872 | 1 ⊢ (𝐾 ∈ 𝑉 → ∧ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (𝐺‘{𝑥, 𝑦}))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cpr 4569 class class class wbr 5066 dom cdm 5555 Fun wfun 6349 ‘cfv 6355 {coprab 7157 glbcglb 17553 meetcmee 17555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-oprab 7160 df-glb 17585 df-meet 17587 |
This theorem is referenced by: meetdm 17627 meetval 17629 |
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