Theorem ordtbaslem 22692

Description: Lemma for ordtbas 22696. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
ordtval.1	⊢ 𝑋 = dom 𝑅
ordtval.2	⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})

Assertion

Ref	Expression
ordtbaslem	⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem ordtbaslem

Dummy variables 𝑎 𝑏 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anrot 1101	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
2		ordtval.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = dom 𝑅
3	2	tsrlemax 18539	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
4	1, 3	sylan2br 596	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
5	4	3exp2 1355	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋 → (𝑏 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏))))))
6	5	imp42 428	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
7	6	notbid 318	. . . . . . . . 9 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ ¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
8		ioran 983	. . . . . . . . 9 ⊢ (¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏))
9	7, 8	bitrdi 287	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)))
10	9	rabbidva 3440	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
11		ifcl 4574	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
12	11	ancoms 460	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
13		dmexg 7894	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
14	2, 13	eqeltrid 2838	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
15	14	adantr 482	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V)
16		rabexg 5332	. . . . . . . . . 10 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
18	10, 17	eqeltrd 2834	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V)
19		eqid 2733	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
20		breq2 5153	. . . . . . . . . . . 12 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (𝑦𝑅𝑥 ↔ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
21	20	notbid 318	. . . . . . . . . . 11 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
22	21	rabbidv 3441	. . . . . . . . . 10 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)})
23	19, 22	elrnmpt1s 5957	. . . . . . . . 9 ⊢ ((if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
24		ordtval.2	. . . . . . . . 9 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
25	23, 24	eleqtrrdi 2845	. . . . . . . 8 ⊢ ((if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
26	12, 18, 25	syl2an2 685	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
27	10, 26	eqeltrrd 2835	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
28	27	ralrimivva 3201	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
29		rabexg 5332	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
30	14, 29	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
31	30	ralrimivw 3151	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
32		breq2 5153	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎))
33	32	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
34	33	rabbidv 3441	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
35	34	cbvmptv 5262	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
36		ineq1 4206	. . . . . . . . . 10 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
37		inrab 4307	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)}
38	36, 37	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
39	38	eleq1d 2819	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → ((𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
40	39	ralbidv 3178	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
41	35, 40	ralrnmptw 7096	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → (∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
42	31, 41	syl 17	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
43	28, 42	mpbird 257	. . . 4 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴)
44		rabexg 5332	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
45	14, 44	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
46	45	ralrimivw 3151	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
47		breq2 5153	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑏))
48	47	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑏))
49	48	rabbidv 3441	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
50	49	cbvmptv 5262	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
51		ineq2 4207	. . . . . . . 8 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → (𝑧 ∩ 𝑤) = (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
52	51	eleq1d 2819	. . . . . . 7 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → ((𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
53	50, 52	ralrnmptw 7096	. . . . . 6 ⊢ (∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V → (∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
54	46, 53	syl 17	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
55	54	ralbidv 3178	. . . 4 ⊢ (𝑅 ∈ TosetRel → (∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
56	43, 55	mpbird 257	. . 3 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
57	24	raleqi 3324	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
58	24, 57	raleqbii 3339	. . 3 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
59	56, 58	sylibr 233	. 2 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴)
60	14	pwexd 5378	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝒫 𝑋 ∈ V)
61		ssrab2 4078	. . . . . . . 8 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
62	14	adantr 482	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
63		elpw2g 5345	. . . . . . . . 9 ⊢ (𝑋 ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
64	62, 63	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
65	61, 64	mpbiri 258	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
66	65	fmpttd 7115	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
67	66	frnd 6726	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
68	24, 67	eqsstrid 4031	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝐴 ⊆ 𝒫 𝑋)
69	60, 68	ssexd 5325	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V)
70		inficl 9420	. . 3 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
71	69, 70	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
72	59, 71	mpbid 231	1 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  ifcif 4529  𝒫 cpw 4603   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ran crn 5678  ‘cfv 6544  ficfi 9405   TosetRel ctsr 18518

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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-fin 8943  df-fi 9406  df-ps 18519  df-tsr 18520

This theorem is referenced by:  ordtbas2  22695