Theorem ipoval 18457

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 3462	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		ipoval.i	. . 3 ⊢ 𝐼 = (toInc‘𝐹)
3		vex 3445	. . . . . . . 8 ⊢ 𝑓 ∈ V
4	3, 3	xpex 7700	. . . . . . 7 ⊢ (𝑓 × 𝑓) ∈ V
5		simpl 482	. . . . . . . . . 10 ⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → {𝑥, 𝑦} ⊆ 𝑓)
6		vex 3445	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
7		vex 3445	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
8	6, 7	prss 4777	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓)
9	5, 8	sylibr 234	. . . . . . . . 9 ⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓))
10	9	ssopab2i 5499	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
11		df-xp 5631	. . . . . . . 8 ⊢ (𝑓 × 𝑓) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)}
12	10, 11	sseqtrri 3984	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝑓 × 𝑓)
13	4, 12	ssexi 5268	. . . . . 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 3961	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹))
16	15	anbi1d 632	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)))
17	16	opabbidv 5165	. . . . . 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 2790	. . . . 5 ⊢ (𝑓 = 𝐹 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = ≤ )
20		simpl 482	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑓 = 𝐹)
21	20	opeq2d 4837	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(Base‘ndx), 𝑓⟩ = ⟨(Base‘ndx), 𝐹⟩)
22		simpr 484	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑜 = ≤ )
23	22	fveq2d 6839	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (ordTop‘𝑜) = (ordTop‘ ≤ ))
24	23	opeq2d 4837	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩ = ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩)
25	21, 24	preq12d 4699	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → {⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩})
26	22	opeq2d 4837	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(le‘ndx), 𝑜⟩ = ⟨(le‘ndx), ≤ ⟩)
27		id 22	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
28		rabeq 3414	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
29	28	unieqd 4877	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})
30	27, 29	mpteq12dv 5186	. . . . . . . . 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 4837	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩ = ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩)
33	26, 32	preq12d 4699	. . . . . 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 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) = ∅})⟩}))
35	14, 19, 34	csbied2 3887	. . . 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 18455	. . . 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 5383	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∈ V
38		prex 5383	. . . . 5 ⊢ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩} ∈ V
39	37, 38	unex 7691	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}) ∈ V
40	35, 36, 39	fvmpt 6942	. . 3 ⊢ (𝐹 ∈ V → (toInc‘𝐹) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
41	2, 40	eqtrid 2784	. 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 395   = wceq 1542   ∈ wcel 2114  {crab 3400  Vcvv 3441  ⦋csb 3850   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  {cpr 4583  ⟨cop 4587  ∪ cuni 4864  {copab 5161   ↦ cmpt 5180   × cxp 5623  ‘cfv 6493  ndxcnx 17124  Basecbs 17140  TopSetcts 17187  lecple 17188  occoc 17189  ordTopcordt 17424  toInccipo 18454

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ipo 18455

This theorem is referenced by:  ipobas  18458  ipolerval  18459  ipotset  18460