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Theorem ipoval 18588
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 3499 . 2 (𝐹𝑉𝐹 ∈ V)
2 ipoval.i . . 3 𝐼 = (toInc‘𝐹)
3 vex 3482 . . . . . . . 8 𝑓 ∈ V
43, 3xpex 7772 . . . . . . 7 (𝑓 × 𝑓) ∈ V
5 simpl 482 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6 vex 3482 . . . . . . . . . . 11 𝑥 ∈ V
7 vex 3482 . . . . . . . . . . 11 𝑦 ∈ V
86, 7prss 4825 . . . . . . . . . 10 ((𝑥𝑓𝑦𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
95, 8sylibr 234 . . . . . . . . 9 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → (𝑥𝑓𝑦𝑓))
109ssopab2i 5560 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
11 df-xp 5695 . . . . . . . 8 (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
1210, 11sseqtrri 4033 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ (𝑓 × 𝑓)
134, 12ssexi 5328 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V
1413a1i 11 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V)
15 sseq2 4022 . . . . . . . 8 (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
1615anbi1d 631 . . . . . . 7 (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)))
1716opabbidv 5214 . . . . . 6 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)})
18 ipoval.l . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)}
1917, 18eqtr4di 2793 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = )
20 simpl 482 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → 𝑓 = 𝐹)
2120opeq2d 4885 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22 simpr 484 . . . . . . . . 9 ((𝑓 = 𝐹𝑜 = ) → 𝑜 = )
2322fveq2d 6911 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (ordTop‘𝑜) = (ordTop‘ ))
2423opeq2d 4885 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ )⟩)
2521, 24preq12d 4746 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩})
2622opeq2d 4885 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ⟩)
27 id 22 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
28 rabeq 3448 . . . . . . . . . . 11 (𝑓 = 𝐹 → {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
2928unieqd 4925 . . . . . . . . . 10 (𝑓 = 𝐹 {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
3027, 29mpteq12dv 5239 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3130adantr 480 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3231opeq2d 4885 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩)
3326, 32preq12d 4746 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩} = {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩})
3425, 33uneq12d 4179 . . . . 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 3948 . . . 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 18586 . . . 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 5443 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∈ V
38 prex 5443 . . . . 5 {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩} ∈ V
3937, 38unex 7763 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}) ∈ V
4035, 36, 39fvmpt 7016 . . 3 (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
412, 40eqtrid 2787 . 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 1537  wcel 2106  {crab 3433  Vcvv 3478  csb 3908  cun 3961  cin 3962  wss 3963  c0 4339  {cpr 4633  cop 4637   cuni 4912  {copab 5210  cmpt 5231   × cxp 5687  cfv 6563  ndxcnx 17227  Basecbs 17245  TopSetcts 17304  lecple 17305  occoc 17306  ordTopcordt 17546  toInccipo 18585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ipo 18586
This theorem is referenced by:  ipobas  18589  ipolerval  18590  ipotset  18591
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