Proof of Theorem hartogslem1
Step | Hyp | Ref
| Expression |
1 | | hartogslem.2 |
. . . . 5
⊢ 𝐹 = {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
2 | 1 | dmeqi 5899 |
. . . 4
⊢ dom 𝐹 = dom {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
3 | | dmopab 5910 |
. . . 4
⊢ dom
{〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑟 ∣ ∃𝑦(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
4 | 2, 3 | eqtri 2761 |
. . 3
⊢ dom 𝐹 = {𝑟 ∣ ∃𝑦(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
5 | | simp3 1139 |
. . . . . . . 8
⊢ ((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) |
6 | | simp1 1137 |
. . . . . . . . 9
⊢ ((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → dom 𝑟 ⊆ 𝐴) |
7 | | xpss12 5687 |
. . . . . . . . 9
⊢ ((dom
𝑟 ⊆ 𝐴 ∧ dom 𝑟 ⊆ 𝐴) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴)) |
8 | 6, 6, 7 | syl2anc 585 |
. . . . . . . 8
⊢ ((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴)) |
9 | 5, 8 | sstrd 3990 |
. . . . . . 7
⊢ ((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (𝐴 × 𝐴)) |
10 | | velpw 4603 |
. . . . . . 7
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴)) |
11 | 9, 10 | sylibr 233 |
. . . . . 6
⊢ ((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴)) |
12 | 11 | ad2antrr 725 |
. . . . 5
⊢ ((((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴)) |
13 | 12 | exlimiv 1934 |
. . . 4
⊢
(∃𝑦(((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴)) |
14 | 13 | abssi 4065 |
. . 3
⊢ {𝑟 ∣ ∃𝑦(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝐴 × 𝐴) |
15 | 4, 14 | eqsstri 4014 |
. 2
⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) |
16 | | funopab4 6577 |
. . 3
⊢ Fun
{〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
17 | 1 | funeqi 6561 |
. . 3
⊢ (Fun
𝐹 ↔ Fun {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}) |
18 | 16, 17 | mpbir 230 |
. 2
⊢ Fun 𝐹 |
19 | 1 | rneqi 5931 |
. . . 4
⊢ ran 𝐹 = ran {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
20 | | rnopab 5948 |
. . . 4
⊢ ran
{〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑦 ∣ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
21 | 19, 20 | eqtri 2761 |
. . 3
⊢ ran 𝐹 = {𝑦 ∣ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
22 | | breq1 5147 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴)) |
23 | 22 | elrab 3681 |
. . . . 5
⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)) |
24 | | f1f 6777 |
. . . . . . . . . . . . 13
⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦⟶𝐴) |
25 | 24 | adantl 483 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓:𝑦⟶𝐴) |
26 | 25 | frnd 6715 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ran 𝑓 ⊆ 𝐴) |
27 | | resss 6001 |
. . . . . . . . . . . . . 14
⊢ ( I
↾ ran 𝑓) ⊆
I |
28 | | ssun2 4171 |
. . . . . . . . . . . . . 14
⊢ I
⊆ (𝑅 ∪ I
) |
29 | 27, 28 | sstri 3989 |
. . . . . . . . . . . . 13
⊢ ( I
↾ ran 𝑓) ⊆
(𝑅 ∪ I
) |
30 | | idssxp 6041 |
. . . . . . . . . . . . 13
⊢ ( I
↾ ran 𝑓) ⊆ (ran
𝑓 × ran 𝑓) |
31 | 29, 30 | ssini 4229 |
. . . . . . . . . . . 12
⊢ ( I
↾ ran 𝑓) ⊆
((𝑅 ∪ I ) ∩ (ran
𝑓 × ran 𝑓)) |
32 | 31 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))) |
33 | | inss2 4227 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓) |
34 | 33 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) |
35 | 26, 32, 34 | 3jca 1129 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓))) |
36 | | eloni 6366 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ On → Ord 𝑦) |
37 | | ordwe 6369 |
. . . . . . . . . . . . . 14
⊢ (Ord
𝑦 → E We 𝑦) |
38 | 36, 37 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ On → E We 𝑦) |
39 | 38 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → E We 𝑦) |
40 | | f1f1orn 6834 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:𝑦–1-1→𝐴 → 𝑓:𝑦–1-1-onto→ran
𝑓) |
41 | 40 | adantl 483 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓:𝑦–1-1-onto→ran
𝑓) |
42 | | hartogslem.3 |
. . . . . . . . . . . . . . 15
⊢ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} |
43 | | f1oiso 7335 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:𝑦–1-1-onto→ran
𝑓 ∧ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)}) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓)) |
44 | 41, 42, 43 | sylancl 587 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓)) |
45 | | isores2 7317 |
. . . . . . . . . . . . . 14
⊢ (𝑓 Isom E , 𝑅 (𝑦, ran 𝑓) ↔ 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓)) |
46 | 44, 45 | sylib 217 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓)) |
47 | | isowe 7333 |
. . . . . . . . . . . . 13
⊢ (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) → ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓)) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓)) |
49 | 39, 48 | mpbid 231 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓) |
50 | | weso 5663 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓) |
52 | | inss2 4227 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓) |
53 | 52 | brel 5736 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → (𝑥 ∈ ran 𝑓 ∧ 𝑥 ∈ ran 𝑓)) |
54 | 53 | simpld 496 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → 𝑥 ∈ ran 𝑓) |
55 | | sonr 5607 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓 ∧ 𝑥 ∈ ran 𝑓) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) |
56 | 51, 54, 55 | syl2an 597 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) ∧ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) |
57 | 56 | pm2.01da 798 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) |
58 | 57 | alrimiv 1931 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) |
59 | | intirr 6111 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) |
60 | 58, 59 | sylibr 233 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅) |
61 | | disj3 4451 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )) |
62 | 60, 61 | sylib 217 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )) |
63 | | weeq1 5660 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)) |
65 | 49, 64 | mpbid 231 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓) |
66 | 36 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → Ord 𝑦) |
67 | | isoeq3 7303 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓))) |
68 | 62, 67 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓))) |
69 | 46, 68 | mpbid 231 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)) |
70 | | vex 3479 |
. . . . . . . . . . . . . . 15
⊢ 𝑓 ∈ V |
71 | 70 | rnex 7890 |
. . . . . . . . . . . . . 14
⊢ ran 𝑓 ∈ V |
72 | | exse 5635 |
. . . . . . . . . . . . . 14
⊢ (ran
𝑓 ∈ V → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓) |
73 | 71, 72 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓 |
74 | | eqid 2733 |
. . . . . . . . . . . . . 14
⊢
OrdIso(((𝑅 ∩
(ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) |
75 | 74 | oieu 9521 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓 ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓) → ((Ord 𝑦 ∧ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)))) |
76 | 65, 73, 75 | sylancl 587 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ((Ord 𝑦 ∧ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)))) |
77 | 66, 69, 76 | mpbi2and 711 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))) |
78 | 77 | simpld 496 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) |
79 | 71, 71 | xpex 7727 |
. . . . . . . . . . . 12
⊢ (ran
𝑓 × ran 𝑓) ∈ V |
80 | 79 | inex2 5314 |
. . . . . . . . . . 11
⊢ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∈ V |
81 | | sseq1 4005 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ⊆ (ran 𝑓 × ran 𝑓) ↔ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓))) |
82 | 33, 81 | mpbiri 258 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 ⊆ (ran 𝑓 × ran 𝑓)) |
83 | | dmss 5897 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 ⊆ (ran 𝑓 × ran 𝑓) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓)) |
84 | 82, 83 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓)) |
85 | | dmxpid 5924 |
. . . . . . . . . . . . . . . . 17
⊢ dom (ran
𝑓 × ran 𝑓) = ran 𝑓 |
86 | 84, 85 | sseqtrdi 4030 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ ran 𝑓) |
87 | | dmresi 6044 |
. . . . . . . . . . . . . . . . 17
⊢ dom ( I
↾ ran 𝑓) = ran 𝑓 |
88 | | sseq2 4006 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (( I ↾ ran 𝑓) ⊆ 𝑟 ↔ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)))) |
89 | 31, 88 | mpbiri 258 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ ran 𝑓) ⊆ 𝑟) |
90 | | dmss 5897 |
. . . . . . . . . . . . . . . . . 18
⊢ (( I
↾ ran 𝑓) ⊆
𝑟 → dom ( I ↾
ran 𝑓) ⊆ dom 𝑟) |
91 | 89, 90 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom ( I ↾ ran 𝑓) ⊆ dom 𝑟) |
92 | 87, 91 | eqsstrrid 4029 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ran 𝑓 ⊆ dom 𝑟) |
93 | 86, 92 | eqssd 3997 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 = ran 𝑓) |
94 | 93 | sseq1d 4011 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟 ⊆ 𝐴 ↔ ran 𝑓 ⊆ 𝐴)) |
95 | 93 | reseq2d 5976 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ dom 𝑟) = ( I ↾ ran 𝑓)) |
96 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))) |
97 | 95, 96 | sseq12d 4013 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)))) |
98 | 93 | sqxpeqd 5704 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟 × dom 𝑟) = (ran 𝑓 × ran 𝑓)) |
99 | 96, 98 | sseq12d 4013 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ⊆ (dom 𝑟 × dom 𝑟) ↔ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓))) |
100 | 94, 97, 99 | 3anbi123d 1437 |
. . . . . . . . . . . . 13
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ↔ (ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)))) |
101 | | difeq1 4113 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I )) |
102 | | difun2 4478 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∪ I ) ∖ I ) = (𝑅 ∖ I ) |
103 | 102 | ineq1i 4206 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∪ I ) ∖ I ) ∩
(ran 𝑓 × ran 𝑓)) = ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) |
104 | | indif1 4269 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∪ I ) ∖ I ) ∩
(ran 𝑓 × ran 𝑓)) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) |
105 | | indif1 4269 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) |
106 | 103, 104,
105 | 3eqtr3i 2769 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) |
107 | 101, 106 | eqtrdi 2789 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )) |
108 | | weeq1 5660 |
. . . . . . . . . . . . . . 15
⊢ ((𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → ((𝑟 ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟)) |
109 | 107, 108 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((𝑟 ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟)) |
110 | | weeq2 5661 |
. . . . . . . . . . . . . . 15
⊢ (dom
𝑟 = ran 𝑓 → (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)) |
111 | 93, 110 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)) |
112 | 109, 111 | bitrd 279 |
. . . . . . . . . . . . 13
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((𝑟 ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)) |
113 | 100, 112 | anbi12d 632 |
. . . . . . . . . . . 12
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ↔ ((ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))) |
114 | | oieq1 9494 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟)) |
115 | 107, 114 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟)) |
116 | | oieq2 9495 |
. . . . . . . . . . . . . . . 16
⊢ (dom
𝑟 = ran 𝑓 → OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) |
117 | 93, 116 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) |
118 | 115, 117 | eqtrd 2773 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) |
119 | 118 | dmeqd 5900 |
. . . . . . . . . . . . 13
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) |
120 | 119 | eqeq2d 2744 |
. . . . . . . . . . . 12
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟) ↔ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))) |
121 | 113, 120 | anbi12d 632 |
. . . . . . . . . . 11
⊢ (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) ↔ (((ran 𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)))) |
122 | 80, 121 | spcev 3595 |
. . . . . . . . . 10
⊢ ((((ran
𝑓 ⊆ 𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) |
123 | 35, 65, 78, 122 | syl21anc 837 |
. . . . . . . . 9
⊢ ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) |
124 | 123 | ex 414 |
. . . . . . . 8
⊢ (𝑦 ∈ On → (𝑓:𝑦–1-1→𝐴 → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))) |
125 | 124 | exlimdv 1937 |
. . . . . . 7
⊢ (𝑦 ∈ On → (∃𝑓 𝑓:𝑦–1-1→𝐴 → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))) |
126 | | brdomi 8942 |
. . . . . . 7
⊢ (𝑦 ≼ 𝐴 → ∃𝑓 𝑓:𝑦–1-1→𝐴) |
127 | 125, 126 | impel 507 |
. . . . . 6
⊢ ((𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) → ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) |
128 | | simpr 486 |
. . . . . . . . . . 11
⊢ ((((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) |
129 | | vex 3479 |
. . . . . . . . . . . . 13
⊢ 𝑟 ∈ V |
130 | 129 | dmex 7889 |
. . . . . . . . . . . 12
⊢ dom 𝑟 ∈ V |
131 | | eqid 2733 |
. . . . . . . . . . . . 13
⊢
OrdIso((𝑟 ∖ I
), dom 𝑟) = OrdIso((𝑟 ∖ I ), dom 𝑟) |
132 | 131 | oion 9518 |
. . . . . . . . . . . 12
⊢ (dom
𝑟 ∈ V → dom
OrdIso((𝑟 ∖ I ), dom
𝑟) ∈
On) |
133 | 130, 132 | ax-mp 5 |
. . . . . . . . . . 11
⊢ dom
OrdIso((𝑟 ∖ I ), dom
𝑟) ∈
On |
134 | 128, 133 | eqeltrdi 2842 |
. . . . . . . . . 10
⊢ ((((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 ∈ On) |
135 | 134 | adantl 483 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → 𝑦 ∈ On) |
136 | | simplr 768 |
. . . . . . . . . . . 12
⊢ ((((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑟 ∖ I ) We dom 𝑟) |
137 | 131 | oien 9520 |
. . . . . . . . . . . 12
⊢ ((dom
𝑟 ∈ V ∧ (𝑟 ∖ I ) We dom 𝑟) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟) |
138 | 130, 136,
137 | sylancr 588 |
. . . . . . . . . . 11
⊢ ((((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟) |
139 | 128, 138 | eqbrtrd 5166 |
. . . . . . . . . 10
⊢ ((((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 ≈ dom 𝑟) |
140 | | ssdomg 8984 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑉 → (dom 𝑟 ⊆ 𝐴 → dom 𝑟 ≼ 𝐴)) |
141 | | simpll1 1213 |
. . . . . . . . . . 11
⊢ ((((dom
𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → dom 𝑟 ⊆ 𝐴) |
142 | 140, 141 | impel 507 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → dom 𝑟 ≼ 𝐴) |
143 | | endomtr 8996 |
. . . . . . . . . 10
⊢ ((𝑦 ≈ dom 𝑟 ∧ dom 𝑟 ≼ 𝐴) → 𝑦 ≼ 𝐴) |
144 | 139, 142,
143 | syl2an2 685 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → 𝑦 ≼ 𝐴) |
145 | 135, 144 | jca 513 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)) |
146 | 145 | ex 414 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → ((((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))) |
147 | 146 | exlimdv 1937 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → (∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))) |
148 | 127, 147 | impbid2 225 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ On ∧ 𝑦 ≼ 𝐴) ↔ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))) |
149 | 23, 148 | bitrid 283 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))) |
150 | 149 | eqabdv 2868 |
. . 3
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} = {𝑦 ∣ ∃𝑟(((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}) |
151 | 21, 150 | eqtr4id 2792 |
. 2
⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) |
152 | 15, 18, 151 | 3pm3.2i 1340 |
1
⊢ (dom
𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) |