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Theorem ipoval 18576
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 3500 . 2 (𝐹𝑉𝐹 ∈ V)
2 ipoval.i . . 3 𝐼 = (toInc‘𝐹)
3 vex 3483 . . . . . . . 8 𝑓 ∈ V
43, 3xpex 7774 . . . . . . 7 (𝑓 × 𝑓) ∈ V
5 simpl 482 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6 vex 3483 . . . . . . . . . . 11 𝑥 ∈ V
7 vex 3483 . . . . . . . . . . 11 𝑦 ∈ V
86, 7prss 4819 . . . . . . . . . 10 ((𝑥𝑓𝑦𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
95, 8sylibr 234 . . . . . . . . 9 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → (𝑥𝑓𝑦𝑓))
109ssopab2i 5554 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
11 df-xp 5690 . . . . . . . 8 (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
1210, 11sseqtrri 4032 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ (𝑓 × 𝑓)
134, 12ssexi 5321 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V
1413a1i 11 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V)
15 sseq2 4009 . . . . . . . 8 (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
1615anbi1d 631 . . . . . . 7 (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)))
1716opabbidv 5208 . . . . . 6 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)})
18 ipoval.l . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)}
1917, 18eqtr4di 2794 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = )
20 simpl 482 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → 𝑓 = 𝐹)
2120opeq2d 4879 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22 simpr 484 . . . . . . . . 9 ((𝑓 = 𝐹𝑜 = ) → 𝑜 = )
2322fveq2d 6909 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (ordTop‘𝑜) = (ordTop‘ ))
2423opeq2d 4879 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ )⟩)
2521, 24preq12d 4740 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩})
2622opeq2d 4879 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ⟩)
27 id 22 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
28 rabeq 3450 . . . . . . . . . . 11 (𝑓 = 𝐹 → {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
2928unieqd 4919 . . . . . . . . . 10 (𝑓 = 𝐹 {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
3027, 29mpteq12dv 5232 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3130adantr 480 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3231opeq2d 4879 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩)
3326, 32preq12d 4740 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩} = {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩})
3425, 33uneq12d 4168 . . . . 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 3935 . . . 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 18574 . . . 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 5436 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∈ V
38 prex 5436 . . . . 5 {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩} ∈ V
3937, 38unex 7765 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}) ∈ V
4035, 36, 39fvmpt 7015 . . 3 (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
412, 40eqtrid 2788 . 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 395   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  csb 3898  cun 3948  cin 3949  wss 3950  c0 4332  {cpr 4627  cop 4631   cuni 4906  {copab 5204  cmpt 5224   × cxp 5682  cfv 6560  ndxcnx 17231  Basecbs 17248  TopSetcts 17304  lecple 17305  occoc 17306  ordTopcordt 17545  toInccipo 18573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ipo 18574
This theorem is referenced by:  ipobas  18577  ipolerval  18578  ipotset  18579
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