Theorem ordtbaslem 23326

Description: Lemma for ordtbas 23330. 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 1117	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
2		ordtval.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = dom 𝑅
3	2	tsrlemax 18643	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
4	1, 3	sylan2br 606	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
5	4	3exp2 1373	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋 → (𝑏 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏))))))
6	5	imp42 431	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
7	6	notbid 321	. . . . . . . . 9 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ ¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
8		ioran 999	. . . . . . . . 9 ⊢ (¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏))
9	7, 8	bitrdi 290	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)))
10	9	rabbidva 3422	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
11		ifcl 4534	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
12	11	ancoms 463	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
13		dmexg 7899	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
14	2, 13	eqeltrid 2867	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
15	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V)
16		rabexg 5309	. . . . . . . . . 10 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
17	15, 16	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
18	10, 17	eqeltrd 2863	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V)
19		eqid 2763	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
20		breq2 5114	. . . . . . . . . . . 12 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (𝑦𝑅𝑥 ↔ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
21	20	notbid 321	. . . . . . . . . . 11 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
22	21	rabbidv 3423	. . . . . . . . . 10 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)})
23	19, 22	elrnmpt1s 5951	. . . . . . . . 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 2874	. . . . . . . 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 698	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
27	10, 26	eqeltrrd 2864	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
28	27	ralrimivva 3208	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
29		rabexg 5309	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
30	14, 29	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
31	30	ralrimivw 3161	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
32		breq2 5114	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎))
33	32	notbid 321	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
34	33	rabbidv 3423	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
35	34	cbvmptv 5216	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
36		ineq1 4167	. . . . . . . . . 10 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
37		inrab 4270	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)}
38	36, 37	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
39	38	eleq1d 2848	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → ((𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
40	39	ralbidv 3188	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
41	35, 40	ralrnmptw 7091	. . . . . 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 18	. . . . 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 260	. . . 4 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴)
44		rabexg 5309	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
45	14, 44	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
46	45	ralrimivw 3161	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
47		breq2 5114	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑏))
48	47	notbid 321	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑏))
49	48	rabbidv 3423	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
50	49	cbvmptv 5216	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
51		ineq2 4168	. . . . . . . 8 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → (𝑧 ∩ 𝑤) = (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
52	51	eleq1d 2848	. . . . . . 7 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → ((𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
53	50, 52	ralrnmptw 7091	. . . . . 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 18	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
55	54	ralbidv 3188	. . . 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 260	. . 3 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
57	24	raleqi 3321	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
58	24, 57	raleqbii 3336	. . 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 237	. 2 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴)
60	14	pwexd 5352	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝒫 𝑋 ∈ V)
61		ssrab2 4035	. . . . . . . 8 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
62	14	adantr 485	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
63		elpw2g 5305	. . . . . . . . 9 ⊢ (𝑋 ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
64	62, 63	syl 18	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
65	61, 64	mpbiri 261	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
66	65	fmpttd 7112	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
67	66	frnd 6716	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
68	24, 67	eqsstrid 3976	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝐴 ⊆ 𝒫 𝑋)
69	60, 68	ssexd 5296	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V)
70		inficl 9386	. . 3 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
71	69, 70	syl 18	. 2 ⊢ (𝑅 ∈ TosetRel → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
72	59, 71	mpbid 235	1 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ifcif 4488  𝒫 cpw 4563   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5663  ran crn 5664  ‘cfv 6538  ficfi 9371   TosetRel ctsr 18622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7864  df-1o 8454  df-2o 8455  df-en 8945  df-fin 8948  df-fi 9372  df-ps 18623  df-tsr 18624

This theorem is referenced by:  ordtbas2  23329