Theorem ipoval 17990

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 3416	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		ipoval.i	. . 3 ⊢ 𝐼 = (toInc‘𝐹)
3		vex 3402	. . . . . . . 8 ⊢ 𝑓 ∈ V
4	3, 3	xpex 7516	. . . . . . 7 ⊢ (𝑓 × 𝑓) ∈ V
5		simpl 486	. . . . . . . . . 10 ⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6		vex 3402	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
7		vex 3402	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	6, 7	prss 4719	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
9	5, 8	sylibr 237	. . . . . . . . 9 ⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓))
10	9	ssopab2i 5416	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
11		df-xp 5542	. . . . . . . 8 ⊢ (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
12	10, 11	sseqtrri 3924	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝑓 × 𝑓)
13	4, 12	ssexi 5200	. . . . . 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 3913	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
16	15	anbi1d 633	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)))
17	16	opabbidv 5105	. . . . . 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	eqtr4di 2789	. . . . 5 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = ≤ )
20		simpl 486	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑓 = 𝐹)
21	20	opeq2d 4777	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22		simpr 488	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑜 = ≤ )
23	22	fveq2d 6699	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (ordTop‘𝑜) = (ordTop‘ ≤ ))
24	23	opeq2d 4777	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩)
25	21, 24	preq12d 4643	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩})
26	22	opeq2d 4777	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ≤ ⟩)
27		id 22	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
28		rabeq 3384	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
29	28	unieqd 4819	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
30	27, 29	mpteq12dv 5125	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}))
31	30	adantr 484	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}))
32	31	opeq2d 4777	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩)
33	26, 32	preq12d 4643	. . . . . 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 4064	. . . . 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 3838	. . . 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 17988	. . . 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 5310	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∈ V
38		prex 5310	. . . . 5 ⊢ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩} ∈ V
39	37, 38	unex 7509	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}) ∈ V
40	35, 36, 39	fvmpt 6796	. . 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 2783	. 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 399   = wceq 1543   ∈ wcel 2112  {crab 3055  Vcvv 3398  ⦋csb 3798   ∪ cun 3851   ∩ cin 3852   ⊆ wss 3853  ∅c0 4223  {cpr 4529  ⟨cop 4533  ∪ cuni 4805  {copab 5101   ↦ cmpt 5120   × cxp 5534  ‘cfv 6358  ndxcnx 16663  Basecbs 16666  TopSetcts 16755  lecple 16756  occoc 16757  ordTopcordt 16958  toInccipo 17987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6316  df-fun 6360  df-fv 6366  df-ipo 17988

This theorem is referenced by:  ipobas  17991  ipolerval  17992  ipotset  17993