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Mirrors > Home > ILE Home > Th. List > iordsmo | GIF version |
Description: The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Ref | Expression |
---|---|
iordsmo.1 | ⊢ Ord 𝐴 |
Ref | Expression |
---|---|
iordsmo | ⊢ Smo ( I ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresi 5210 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
2 | rnresi 4866 | . . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴 | |
3 | iordsmo.1 | . . . . 5 ⊢ Ord 𝐴 | |
4 | ordsson 4378 | . . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐴 ⊆ On |
6 | 2, 5 | eqsstri 3099 | . . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On |
7 | df-f 5097 | . . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On)) | |
8 | 1, 6, 7 | mpbir2an 911 | . 2 ⊢ ( I ↾ 𝐴):𝐴⟶On |
9 | fvresi 5581 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
10 | 9 | adantr 274 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥) |
11 | fvresi 5581 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦) | |
12 | 11 | adantl 275 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦) |
13 | 10, 12 | eleq12d 2188 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦)) |
14 | 13 | biimprd 157 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦))) |
15 | dmresi 4844 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
16 | 8, 3, 14, 15 | issmo 6153 | 1 ⊢ Smo ( I ↾ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 I cid 4180 Ord word 4254 Oncon0 4255 ran crn 4510 ↾ cres 4511 Fn wfn 5088 ⟶wf 5089 ‘cfv 5093 Smo wsmo 6150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-smo 6151 |
This theorem is referenced by: smo0 6163 |
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