Theorem ipoval 16977

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} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}

Assertion

Ref	Expression
ipoval	⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))

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.

Step	Hyp	Ref	Expression
1		elex 3185	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		ipoval.i	. . 3 ⊢ 𝐼 = (toInc‘𝐹)
3		vex 3176	. . . . . . . 8 ⊢ 𝑓 ∈ V
4	3, 3	xpex 6860	. . . . . . 7 ⊢ (𝑓 × 𝑓) ∈ V
5		simpl 472	. . . . . . . . . 10 ⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6		vex 3176	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
7		vex 3176	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	6, 7	prss 4291	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
9	5, 8	sylibr 223	. . . . . . . . 9 ⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓))
10	9	ssopab2i 4928	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
11		df-xp 5044	. . . . . . . 8 ⊢ (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
12	10, 11	sseqtr4i 3601	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝑓 × 𝑓)
13	4, 12	ssexi 4731	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ∈ V
14	13	a1i 11	. . . . 5 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ∈ V)
15		sseq2 3590	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
16	15	anbi1d 737	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)))
17	16	opabbidv 4648	. . . . . 6 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})
18		ipoval.l	. . . . . 6 ⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}
19	17, 18	syl6eqr 2662	. . . . 5 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = ≤ )
20		simpl 472	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑓 = 𝐹)
21	20	opeq2d 4347	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22		simpr 476	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑜 = ≤ )
23	22	fveq2d 6107	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (ordTop‘𝑜) = (ordTop‘ ≤ ))
24	23	opeq2d 4347	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩)
25	21, 24	preq12d 4220	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩})
26	22	opeq2d 4347	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ≤ ⟩)
27		id 22	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
28		rabeq 3166	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
29	28	unieqd 4382	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
30	27, 29	mpteq12dv 4663	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}))
32	31	opeq2d 4347	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩)
33	26, 32	preq12d 4220	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩} = {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩})
34	25, 33	uneq12d 3730	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
35	14, 19, 34	csbied2 3527	. . . 4 ⊢ (𝑓 = 𝐹 → ⦋{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
36		df-ipo 16975	. . . 4 ⊢ toInc = (𝑓 ∈ V ↦ ⦋{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
37		prex 4836	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∈ V
38		prex 4836	. . . . 5 ⊢ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩} ∈ V
39	37, 38	unex 6854	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}) ∈ V
40	35, 36, 39	fvmpt 6191	. . 3 ⊢ (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
41	2, 40	syl5eq 2656	. 2 ⊢ (𝐹 ∈ V → 𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
42	1, 41	syl 17	1 ⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⦋csb 3499   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ⟨cop 4131  ∪ cuni 4372  {copab 4642   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  ndxcnx 15692  Basecbs 15695  TopSetcts 15774  lecple 15775  occoc 15776  ordTopcordt 15982  toInccipo 16974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ipo 16975

This theorem is referenced by:  ipobas  16978  ipolerval  16979  ipotset  16980