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Theorem ipoval 17082
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 3201 . 2 (𝐹𝑉𝐹 ∈ V)
2 ipoval.i . . 3 𝐼 = (toInc‘𝐹)
3 vex 3192 . . . . . . . 8 𝑓 ∈ V
43, 3xpex 6922 . . . . . . 7 (𝑓 × 𝑓) ∈ V
5 simpl 473 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6 vex 3192 . . . . . . . . . . 11 𝑥 ∈ V
7 vex 3192 . . . . . . . . . . 11 𝑦 ∈ V
86, 7prss 4324 . . . . . . . . . 10 ((𝑥𝑓𝑦𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
95, 8sylibr 224 . . . . . . . . 9 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → (𝑥𝑓𝑦𝑓))
109ssopab2i 4968 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
11 df-xp 5085 . . . . . . . 8 (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
1210, 11sseqtr4i 3622 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ (𝑓 × 𝑓)
134, 12ssexi 4768 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V
1413a1i 11 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V)
15 sseq2 3611 . . . . . . . 8 (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
1615anbi1d 740 . . . . . . 7 (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)))
1716opabbidv 4683 . . . . . 6 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)})
18 ipoval.l . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)}
1917, 18syl6eqr 2673 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = )
20 simpl 473 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → 𝑓 = 𝐹)
2120opeq2d 4382 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22 simpr 477 . . . . . . . . 9 ((𝑓 = 𝐹𝑜 = ) → 𝑜 = )
2322fveq2d 6157 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (ordTop‘𝑜) = (ordTop‘ ))
2423opeq2d 4382 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ )⟩)
2521, 24preq12d 4251 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩})
2622opeq2d 4382 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ⟩)
27 id 22 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
28 rabeq 3182 . . . . . . . . . . 11 (𝑓 = 𝐹 → {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
2928unieqd 4417 . . . . . . . . . 10 (𝑓 = 𝐹 {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
3027, 29mpteq12dv 4698 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3130adantr 481 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3231opeq2d 4382 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩)
3326, 32preq12d 4251 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩} = {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩})
3425, 33uneq12d 3751 . . . . 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 3546 . . . 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 17080 . . . 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 4875 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∈ V
38 prex 4875 . . . . 5 {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩} ∈ V
3937, 38unex 6916 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}) ∈ V
4035, 36, 39fvmpt 6244 . . 3 (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
412, 40syl5eq 2667 . 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 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  csb 3518  cun 3557  cin 3558  wss 3559  c0 3896  {cpr 4155  cop 4159   cuni 4407  {copab 4677  cmpt 4678   × cxp 5077  cfv 5852  ndxcnx 15785  Basecbs 15788  TopSetcts 15875  lecple 15876  occoc 15877  ordTopcordt 16087  toInccipo 17079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ipo 17080
This theorem is referenced by:  ipobas  17083  ipolerval  17084  ipotset  17085
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