MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ipoval Structured version   Visualization version   GIF version

Theorem ipoval 18258
Description: Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
ipoval.i 𝐼 = (toInc‘𝐹)
ipoval.l = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)}
Assertion
Ref Expression
ipoval (𝐹𝑉𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐼,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem ipoval
Dummy variables 𝑓 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3447 . 2 (𝐹𝑉𝐹 ∈ V)
2 ipoval.i . . 3 𝐼 = (toInc‘𝐹)
3 vex 3433 . . . . . . . 8 𝑓 ∈ V
43, 3xpex 7593 . . . . . . 7 (𝑓 × 𝑓) ∈ V
5 simpl 483 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6 vex 3433 . . . . . . . . . . 11 𝑥 ∈ V
7 vex 3433 . . . . . . . . . . 11 𝑦 ∈ V
86, 7prss 4753 . . . . . . . . . 10 ((𝑥𝑓𝑦𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
95, 8sylibr 233 . . . . . . . . 9 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → (𝑥𝑓𝑦𝑓))
109ssopab2i 5460 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
11 df-xp 5590 . . . . . . . 8 (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
1210, 11sseqtrri 3957 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ (𝑓 × 𝑓)
134, 12ssexi 5244 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V
1413a1i 11 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V)
15 sseq2 3946 . . . . . . . 8 (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
1615anbi1d 630 . . . . . . 7 (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)))
1716opabbidv 5139 . . . . . 6 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)})
18 ipoval.l . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)}
1917, 18eqtr4di 2796 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = )
20 simpl 483 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → 𝑓 = 𝐹)
2120opeq2d 4811 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22 simpr 485 . . . . . . . . 9 ((𝑓 = 𝐹𝑜 = ) → 𝑜 = )
2322fveq2d 6770 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (ordTop‘𝑜) = (ordTop‘ ))
2423opeq2d 4811 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ )⟩)
2521, 24preq12d 4677 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩})
2622opeq2d 4811 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ⟩)
27 id 22 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
28 rabeq 3415 . . . . . . . . . . 11 (𝑓 = 𝐹 → {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
2928unieqd 4853 . . . . . . . . . 10 (𝑓 = 𝐹 {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
3027, 29mpteq12dv 5164 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3130adantr 481 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3231opeq2d 4811 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩)
3326, 32preq12d 4677 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩} = {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩})
3425, 33uneq12d 4097 . . . . 5 ((𝑓 = 𝐹𝑜 = ) → ({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩}) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
3514, 19, 34csbied2 3871 . . . 4 (𝑓 = 𝐹{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} / 𝑜({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩}) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
36 df-ipo 18256 . . . 4 toInc = (𝑓 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} / 𝑜({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩}))
37 prex 5353 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∈ V
38 prex 5353 . . . . 5 {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩} ∈ V
3937, 38unex 7586 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}) ∈ V
4035, 36, 39fvmpt 6867 . . 3 (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
412, 40eqtrid 2790 . 2 (𝐹 ∈ V → 𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
421, 41syl 17 1 (𝐹𝑉𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {crab 3068  Vcvv 3429  csb 3831  cun 3884  cin 3885  wss 3886  c0 4256  {cpr 4563  cop 4567   cuni 4839  {copab 5135  cmpt 5156   × cxp 5582  cfv 6426  ndxcnx 16904  Basecbs 16922  TopSetcts 16978  lecple 16979  occoc 16980  ordTopcordt 17220  toInccipo 18255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434  df-ipo 18256
This theorem is referenced by:  ipobas  18259  ipolerval  18260  ipotset  18261
  Copyright terms: Public domain W3C validator