Theorem ipoval 18592

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 3475	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		ipoval.i	. . 3 ⊢ 𝐼 = (toInc‘𝐹)
3		vex 3458	. . . . . . . 8 ⊢ 𝑓 ∈ V
4	3, 3	xpex 7750	. . . . . . 7 ⊢ (𝑓 × 𝑓) ∈ V
5		vex 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
6		vex 3458	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	5, 6	prss 4785	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
8	7	biranri 510	. . . . . . . . 9 ⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓))
9	8	ssopab2i 5534	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
10		df-xp 5666	. . . . . . . 8 ⊢ (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
11	9, 10	sseqtrri 3985	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝑓 × 𝑓)
12	4, 11	ssexi 5292	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ∈ V
13	12	a1i 11	. . . . 5 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ∈ V)
14		sseq2 3962	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
15	14	anbi1d 642	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)))
16	15	opabbidv 5176	. . . . . 6 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})
17		ipoval.l	. . . . . 6 ⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}
18	16, 17	eqtr4di 2815	. . . . 5 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = ≤ )
19		simpl 487	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑓 = 𝐹)
20	19	opeq2d 4844	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
21		simpr 489	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑜 = ≤ )
22	21	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (ordTop‘𝑜) = (ordTop‘ ≤ ))
23	22	opeq2d 4844	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩)
24	20, 23	preq12d 4706	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩})
25	21	opeq2d 4844	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ≤ ⟩)
26		id 23	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
27		rabeq 3429	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
28	27	unieqd 4884	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
29	26, 28	mpteq12dv 5197	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}))
30	29	adantr 485	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}))
31	30	opeq2d 4844	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩)
32	25, 31	preq12d 4706	. . . . . 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) = ∅})⟩})
33	24, 32	uneq12d 4122	. . . . 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) = ∅})⟩}))
34	13, 18, 33	csbied2 3889	. . . 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) = ∅})⟩}))
35		df-ipo 18590	. . . 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) = ∅})⟩}))
36		prex 5408	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∈ V
37		prex 5408	. . . . 5 ⊢ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩} ∈ V
38	36, 37	unex 7744	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}) ∈ V
39	34, 35, 38	fvmpt 6989	. . 3 ⊢ (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
40	2, 39	eqtrid 2809	. 2 ⊢ (𝐹 ∈ V → 𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
41	1, 40	syl 18	1 ⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {crab 3415  Vcvv 3454  ⦋csb 3852   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {cpr 4590  ⟨cop 4594  ∪ cuni 4871  {copab 5172   ↦ cmpt 5191   × cxp 5658  ‘cfv 6536  ndxcnx 17259  Basecbs 17275  TopSetcts 17322  lecple 17323  occoc 17324  ordTopcordt 17559  toInccipo 18589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ipo 18590

This theorem is used by:  ipobas  18593  ipolerval  18594  ipotset  18595