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Theorem ipoval 17760
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 3462 . 2 (𝐹𝑉𝐹 ∈ V)
2 ipoval.i . . 3 𝐼 = (toInc‘𝐹)
3 vex 3447 . . . . . . . 8 𝑓 ∈ V
43, 3xpex 7460 . . . . . . 7 (𝑓 × 𝑓) ∈ V
5 simpl 486 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6 vex 3447 . . . . . . . . . . 11 𝑥 ∈ V
7 vex 3447 . . . . . . . . . . 11 𝑦 ∈ V
86, 7prss 4716 . . . . . . . . . 10 ((𝑥𝑓𝑦𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
95, 8sylibr 237 . . . . . . . . 9 (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) → (𝑥𝑓𝑦𝑓))
109ssopab2i 5405 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
11 df-xp 5529 . . . . . . . 8 (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑓𝑦𝑓)}
1210, 11sseqtrri 3955 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ⊆ (𝑓 × 𝑓)
134, 12ssexi 5193 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V
1413a1i 11 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} ∈ V)
15 sseq2 3944 . . . . . . . 8 (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
1615anbi1d 632 . . . . . . 7 (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)))
1716opabbidv 5099 . . . . . 6 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)})
18 ipoval.l . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)}
1917, 18eqtr4di 2854 . . . . 5 (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} = )
20 simpl 486 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → 𝑓 = 𝐹)
2120opeq2d 4775 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22 simpr 488 . . . . . . . . 9 ((𝑓 = 𝐹𝑜 = ) → 𝑜 = )
2322fveq2d 6653 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (ordTop‘𝑜) = (ordTop‘ ))
2423opeq2d 4775 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ )⟩)
2521, 24preq12d 4640 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩})
2622opeq2d 4775 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ⟩)
27 id 22 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
28 rabeq 3434 . . . . . . . . . . 11 (𝑓 = 𝐹 → {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
2928unieqd 4817 . . . . . . . . . 10 (𝑓 = 𝐹 {𝑦𝑓 ∣ (𝑦𝑥) = ∅} = {𝑦𝐹 ∣ (𝑦𝑥) = ∅})
3027, 29mpteq12dv 5118 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3130adantr 484 . . . . . . . 8 ((𝑓 = 𝐹𝑜 = ) → (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅}) = (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅}))
3231opeq2d 4775 . . . . . . 7 ((𝑓 = 𝐹𝑜 = ) → ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩)
3326, 32preq12d 4640 . . . . . 6 ((𝑓 = 𝐹𝑜 = ) → {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩} = {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩})
3425, 33uneq12d 4094 . . . . 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 3868 . . . 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 17758 . . . 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 5301 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∈ V
38 prex 5301 . . . . 5 {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩} ∈ V
3937, 38unex 7453 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}) ∈ V
4035, 36, 39fvmpt 6749 . . 3 (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))
412, 40syl5eq 2848 . 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 399   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  csb 3831  cun 3882  cin 3883  wss 3884  c0 4246  {cpr 4530  cop 4534   cuni 4803  {copab 5095  cmpt 5113   × cxp 5521  cfv 6328  ndxcnx 16476  Basecbs 16479  TopSetcts 16567  lecple 16568  occoc 16569  ordTopcordt 16768  toInccipo 17757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ipo 17758
This theorem is referenced by:  ipobas  17761  ipolerval  17762  ipotset  17763
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