![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ipobas | Structured version Visualization version GIF version |
Description: Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.) |
Ref | Expression |
---|---|
ipoval.i | ⊢ 𝐼 = (toInc‘𝐹) |
Ref | Expression |
---|---|
ipobas | ⊢ (𝐹 ∈ 𝑉 → 𝐹 = (Base‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipostr 17621 | . . 3 ⊢ ({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) Struct 〈1, ;11〉 | |
2 | baseid 16399 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
3 | snsspr1 4621 | . . . 4 ⊢ {〈(Base‘ndx), 𝐹〉} ⊆ {〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} | |
4 | ssun1 4037 | . . . 4 ⊢ {〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ⊆ ({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) | |
5 | 3, 4 | sstri 3867 | . . 3 ⊢ {〈(Base‘ndx), 𝐹〉} ⊆ ({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) |
6 | 1, 2, 5 | strfv 16387 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 = (Base‘({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}))) |
7 | ipoval.i | . . . 4 ⊢ 𝐼 = (toInc‘𝐹) | |
8 | eqid 2778 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} | |
9 | 7, 8 | ipoval 17622 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
10 | 9 | fveq2d 6503 | . 2 ⊢ (𝐹 ∈ 𝑉 → (Base‘𝐼) = (Base‘({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}))) |
11 | 6, 10 | eqtr4d 2817 | 1 ⊢ (𝐹 ∈ 𝑉 → 𝐹 = (Base‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {crab 3092 ∪ cun 3827 ∩ cin 3828 ⊆ wss 3829 ∅c0 4178 {csn 4441 {cpr 4443 〈cop 4447 ∪ cuni 4712 {copab 4991 ↦ cmpt 5008 ‘cfv 6188 1c1 10336 ;cdc 11911 ndxcnx 16336 Basecbs 16339 TopSetcts 16427 lecple 16428 occoc 16429 ordTopcordt 16628 toInccipo 17619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-tset 16440 df-ple 16441 df-ocomp 16442 df-ipo 17620 |
This theorem is referenced by: ipopos 17628 isipodrs 17629 ipodrsfi 17631 mrelatglb 17652 mrelatglb0 17653 mrelatlub 17654 mreclatBAD 17655 thlbas 20542 |
Copyright terms: Public domain | W3C validator |