Theorem ordtbaslem 21404

Description: Lemma for ordtbas 21408. 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 1085	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
2		ordtval.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = dom 𝑅
3	2	tsrlemax 17610	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
4	1, 3	sylan2br 588	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
5	4	3exp2 1416	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋 → (𝑏 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏))))))
6	5	imp42 419	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
7	6	notbid 310	. . . . . . . . 9 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ ¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
8		ioran 969	. . . . . . . . 9 ⊢ (¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏))
9	7, 8	syl6bb 279	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)))
10	9	rabbidva 3385	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
11		ifcl 4351	. . . . . . . . . 10 ⊢ ((𝑏 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
12	11	ancoms 452	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
13	12	adantl 475	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
14		dmexg 7377	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
15	2, 14	syl5eqel 2863	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
16	15	adantr 474	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V)
17		rabexg 5050	. . . . . . . . . 10 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
18	16, 17	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ V)
19	10, 18	eqeltrd 2859	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V)
20		eqid 2778	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
21		breq2 4892	. . . . . . . . . . . 12 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (𝑦𝑅𝑥 ↔ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
22	21	notbid 310	. . . . . . . . . . 11 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
23	22	rabbidv 3386	. . . . . . . . . 10 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)})
24	20, 23	elrnmpt1s 5621	. . . . . . . . 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}))
25		ordtval.2	. . . . . . . . 9 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
26	24, 25	syl6eleqr 2870	. . . . . . . 8 ⊢ ((if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
27	13, 19, 26	syl2anc 579	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
28	10, 27	eqeltrrd 2860	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
29	28	ralrimivva 3153	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
30		rabexg 5050	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
31	15, 30	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
32	31	ralrimivw 3149	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
33		breq2 4892	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎))
34	33	notbid 310	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
35	34	rabbidv 3386	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
36	35	cbvmptv 4987	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
37		ineq1 4030	. . . . . . . . . 10 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
38		inrab 4125	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)}
39	37, 38	syl6eq 2830	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
40	39	eleq1d 2844	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → ((𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
41	40	ralbidv 3168	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
42	36, 41	ralrnmpt 6634	. . . . . 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))
43	32, 42	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))
44	29, 43	mpbird 249	. . . 4 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴)
45		rabexg 5050	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
46	15, 45	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
47	46	ralrimivw 3149	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
48		breq2 4892	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑏))
49	48	notbid 310	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑏))
50	49	rabbidv 3386	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
51	50	cbvmptv 4987	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
52		ineq2 4031	. . . . . . . 8 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → (𝑧 ∩ 𝑤) = (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
53	52	eleq1d 2844	. . . . . . 7 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → ((𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
54	51, 53	ralrnmpt 6634	. . . . . 6 ⊢ (∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V → (∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
55	47, 54	syl 17	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
56	55	ralbidv 3168	. . . 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))
57	44, 56	mpbird 249	. . 3 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
58	25	raleqi 3338	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
59	25, 58	raleqbii 3172	. . 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)
60	57, 59	sylibr 226	. 2 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴)
61	15	pwexd 5093	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝒫 𝑋 ∈ V)
62		ssrab2 3908	. . . . . . . 8 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
63	15	adantr 474	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
64		elpw2g 5063	. . . . . . . . 9 ⊢ (𝑋 ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
65	63, 64	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
66	62, 65	mpbiri 250	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
67	66	fmpttd 6651	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
68	67	frnd 6300	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
69	25, 68	syl5eqss 3868	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝐴 ⊆ 𝒫 𝑋)
70	61, 69	ssexd 5044	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V)
71		inficl 8621	. . 3 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
72	70, 71	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
73	60, 72	mpbid 224	1 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  {crab 3094  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792  ifcif 4307  𝒫 cpw 4379   class class class wbr 4888   ↦ cmpt 4967  dom cdm 5357  ran crn 5358  ‘cfv 6137  ficfi 8606   TosetRel ctsr 17589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-fin 8247  df-fi 8607  df-ps 17590  df-tsr 17591

This theorem is referenced by:  ordtbas2  21407