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) |