Theorem ordtbaslem 20802

Description: Lemma for ordtbas 20806. 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 1036	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
2		ordtval.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = dom 𝑅
3	2	tsrlemax 17043	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑦 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
4	1, 3	sylan2br 492	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
5	4	3exp2 1277	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋 → (𝑏 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏))))))
6	5	imp42 618	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
7	6	notbid 307	. . . . . . . . 9 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ ¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏)))
8		ioran 510	. . . . . . . . 9 ⊢ (¬ (𝑦𝑅𝑎 ∨ 𝑦𝑅𝑏) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏))
9	7, 8	syl6bb 275	. . . . . . . 8 ⊢ (((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → (¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎) ↔ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)))
10	9	rabbidva 3163	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
11		ifcl 4080	. . . . . . . . . 10 ⊢ ((𝑏 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
12	11	ancoms 468	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
13	12	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → if(𝑎𝑅𝑏, 𝑏, 𝑎) ∈ 𝑋)
14		dmexg 6989	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
15	2, 14	syl5eqel 2692	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
16	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V)
17		rabexg 4739	. . . . . . . . . 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 2688	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ V)
20		eqid 2610	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
21		breq2 4587	. . . . . . . . . . . 12 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (𝑦𝑅𝑥 ↔ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
22	21	notbid 307	. . . . . . . . . . 11 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)))
23	22	rabbidv 3164	. . . . . . . . . 10 ⊢ (𝑥 = if(𝑎𝑅𝑏, 𝑏, 𝑎) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)})
24	20, 23	elrnmpt1s 5294	. . . . . . . . 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 2699	. . . . . . . 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 691	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅if(𝑎𝑅𝑏, 𝑏, 𝑎)} ∈ 𝐴)
28	10, 27	eqeltrrd 2689	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
29	28	ralrimivva 2954	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴)
30		rabexg 4739	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
31	15, 30	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
32	31	ralrimivw 2950	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
33		breq2 4587	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎))
34	33	notbid 307	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
35	34	rabbidv 3164	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
36	35	cbvmptv 4678	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
37		ineq1 3769	. . . . . . . . . 10 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
38		inrab 3858	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)}
39	37, 38	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)})
40	39	eleq1d 2672	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → ((𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
41	40	ralbidv 2969	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} → (∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴 ↔ ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑦𝑅𝑏)} ∈ 𝐴))
42	36, 41	ralrnmpt 6276	. . . . . 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 246	. . . 4 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑏 ∈ 𝑋 (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴)
45		rabexg 4739	. . . . . . . 8 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
46	15, 45	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
47	46	ralrimivw 2950	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} ∈ V)
48		breq2 4587	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑏))
49	48	notbid 307	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑏))
50	49	rabbidv 3164	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
51	50	cbvmptv 4678	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏})
52		ineq2 3770	. . . . . . . 8 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → (𝑧 ∩ 𝑤) = (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}))
53	52	eleq1d 2672	. . . . . . 7 ⊢ (𝑤 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏} → ((𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (𝑧 ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑏}) ∈ 𝐴))
54	51, 53	ralrnmpt 6276	. . . . . 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 2969	. . . 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 246	. . 3 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
58	25	raleqi 3119	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})(𝑧 ∩ 𝑤) ∈ 𝐴)
59	25, 58	raleqbii 2973	. . 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 223	. 2 ⊢ (𝑅 ∈ TosetRel → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴)
61		pwexg 4776	. . . . 5 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
62	15, 61	syl 17	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝒫 𝑋 ∈ V)
63		ssrab2 3650	. . . . . . . 8 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
64	15	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
65		elpw2g 4754	. . . . . . . . 9 ⊢ (𝑋 ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
66	64, 65	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
67	63, 66	mpbiri 247	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
68	67, 20	fmptd 6292	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
69		frn 5966	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
70	68, 69	syl 17	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
71	25, 70	syl5eqss 3612	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝐴 ⊆ 𝒫 𝑋)
72	62, 71	ssexd 4733	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V)
73		inficl 8214	. . 3 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
74	72, 73	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ∩ 𝑤) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
75	60, 74	mpbid 221	1 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  ficfi 8199   TosetRel ctsr 17022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-ps 17023  df-tsr 17024

This theorem is referenced by:  ordtbas2  20805