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Theorem ipoval 18424
Description: Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
ipoval.i 𝐼 = (toIncβ€˜πΉ)
ipoval.l ≀ = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐹 ∧ π‘₯ βŠ† 𝑦)}
Assertion
Ref Expression
ipoval (𝐹 ∈ 𝑉 β†’ 𝐼 = ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
Distinct variable groups:   π‘₯,𝑦,𝐹   π‘₯,𝐼,𝑦   π‘₯,𝑉,𝑦
Allowed substitution hints:   ≀ (π‘₯,𝑦)

Proof of Theorem ipoval
Dummy variables 𝑓 π‘œ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3462 . 2 (𝐹 ∈ 𝑉 β†’ 𝐹 ∈ V)
2 ipoval.i . . 3 𝐼 = (toIncβ€˜πΉ)
3 vex 3448 . . . . . . . 8 𝑓 ∈ V
43, 3xpex 7688 . . . . . . 7 (𝑓 Γ— 𝑓) ∈ V
5 simpl 484 . . . . . . . . . 10 (({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦) β†’ {π‘₯, 𝑦} βŠ† 𝑓)
6 vex 3448 . . . . . . . . . . 11 π‘₯ ∈ V
7 vex 3448 . . . . . . . . . . 11 𝑦 ∈ V
86, 7prss 4781 . . . . . . . . . 10 ((π‘₯ ∈ 𝑓 ∧ 𝑦 ∈ 𝑓) ↔ {π‘₯, 𝑦} βŠ† 𝑓)
95, 8sylibr 233 . . . . . . . . 9 (({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦) β†’ (π‘₯ ∈ 𝑓 ∧ 𝑦 ∈ 𝑓))
109ssopab2i 5508 . . . . . . . 8 {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} βŠ† {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
11 df-xp 5640 . . . . . . . 8 (𝑓 Γ— 𝑓) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
1210, 11sseqtrri 3982 . . . . . . 7 {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} βŠ† (𝑓 Γ— 𝑓)
134, 12ssexi 5280 . . . . . 6 {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} ∈ V
1413a1i 11 . . . . 5 (𝑓 = 𝐹 β†’ {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} ∈ V)
15 sseq2 3971 . . . . . . . 8 (𝑓 = 𝐹 β†’ ({π‘₯, 𝑦} βŠ† 𝑓 ↔ {π‘₯, 𝑦} βŠ† 𝐹))
1615anbi1d 631 . . . . . . 7 (𝑓 = 𝐹 β†’ (({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦) ↔ ({π‘₯, 𝑦} βŠ† 𝐹 ∧ π‘₯ βŠ† 𝑦)))
1716opabbidv 5172 . . . . . 6 (𝑓 = 𝐹 β†’ {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐹 ∧ π‘₯ βŠ† 𝑦)})
18 ipoval.l . . . . . 6 ≀ = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐹 ∧ π‘₯ βŠ† 𝑦)}
1917, 18eqtr4di 2791 . . . . 5 (𝑓 = 𝐹 β†’ {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} = ≀ )
20 simpl 484 . . . . . . . 8 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ 𝑓 = 𝐹)
2120opeq2d 4838 . . . . . . 7 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ ⟨(Baseβ€˜ndx), π‘“βŸ© = ⟨(Baseβ€˜ndx), 𝐹⟩)
22 simpr 486 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ π‘œ = ≀ )
2322fveq2d 6847 . . . . . . . 8 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ (ordTopβ€˜π‘œ) = (ordTopβ€˜ ≀ ))
2423opeq2d 4838 . . . . . . 7 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ ⟨(TopSetβ€˜ndx), (ordTopβ€˜π‘œ)⟩ = ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩)
2521, 24preq12d 4703 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ {⟨(Baseβ€˜ndx), π‘“βŸ©, ⟨(TopSetβ€˜ndx), (ordTopβ€˜π‘œ)⟩} = {⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩})
2622opeq2d 4838 . . . . . . 7 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ ⟨(leβ€˜ndx), π‘œβŸ© = ⟨(leβ€˜ndx), ≀ ⟩)
27 id 22 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ 𝑓 = 𝐹)
28 rabeq 3420 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…} = {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})
2928unieqd 4880 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…} = βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})
3027, 29mpteq12dv 5197 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…}) = (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…}))
3130adantr 482 . . . . . . . 8 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…}) = (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…}))
3231opeq2d 4838 . . . . . . 7 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩ = ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩)
3326, 32preq12d 4703 . . . . . 6 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ {⟨(leβ€˜ndx), π‘œβŸ©, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩} = {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩})
3425, 33uneq12d 4125 . . . . 5 ((𝑓 = 𝐹 ∧ π‘œ = ≀ ) β†’ ({⟨(Baseβ€˜ndx), π‘“βŸ©, ⟨(TopSetβ€˜ndx), (ordTopβ€˜π‘œ)⟩} βˆͺ {⟨(leβ€˜ndx), π‘œβŸ©, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}) = ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
3514, 19, 34csbied2 3896 . . . 4 (𝑓 = 𝐹 β†’ ⦋{⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} / π‘œβ¦Œ({⟨(Baseβ€˜ndx), π‘“βŸ©, ⟨(TopSetβ€˜ndx), (ordTopβ€˜π‘œ)⟩} βˆͺ {⟨(leβ€˜ndx), π‘œβŸ©, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}) = ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
36 df-ipo 18422 . . . 4 toInc = (𝑓 ∈ V ↦ ⦋{⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝑓 ∧ π‘₯ βŠ† 𝑦)} / π‘œβ¦Œ({⟨(Baseβ€˜ndx), π‘“βŸ©, ⟨(TopSetβ€˜ndx), (ordTopβ€˜π‘œ)⟩} βˆͺ {⟨(leβ€˜ndx), π‘œβŸ©, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝑓 ↦ βˆͺ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
37 prex 5390 . . . . 5 {⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} ∈ V
38 prex 5390 . . . . 5 {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩} ∈ V
3937, 38unex 7681 . . . 4 ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}) ∈ V
4035, 36, 39fvmpt 6949 . . 3 (𝐹 ∈ V β†’ (toIncβ€˜πΉ) = ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
412, 40eqtrid 2785 . 2 (𝐹 ∈ V β†’ 𝐼 = ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
421, 41syl 17 1 (𝐹 ∈ 𝑉 β†’ 𝐼 = ({⟨(Baseβ€˜ndx), 𝐹⟩, ⟨(TopSetβ€˜ndx), (ordTopβ€˜ ≀ )⟩} βˆͺ {⟨(leβ€˜ndx), ≀ ⟩, ⟨(ocβ€˜ndx), (π‘₯ ∈ 𝐹 ↦ βˆͺ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ π‘₯) = βˆ…})⟩}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3444  β¦‹csb 3856   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  {cpr 4589  βŸ¨cop 4593  βˆͺ cuni 4866  {copab 5168   ↦ cmpt 5189   Γ— cxp 5632  β€˜cfv 6497  ndxcnx 17070  Basecbs 17088  TopSetcts 17144  lecple 17145  occoc 17146  ordTopcordt 17386  toInccipo 18421
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ipo 18422
This theorem is referenced by:  ipobas  18425  ipolerval  18426  ipotset  18427
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