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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 5176 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
2 | rnresi 4832 | . . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴 | |
3 | iordsmo.1 | . . . . 5 ⊢ Ord 𝐴 | |
4 | ordsson 4346 | . . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
5 | 3, 4 | ax-mp 7 | . . . 4 ⊢ 𝐴 ⊆ On |
6 | 2, 5 | eqsstri 3079 | . . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On |
7 | df-f 5063 | . . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On)) | |
8 | 1, 6, 7 | mpbir2an 894 | . 2 ⊢ ( I ↾ 𝐴):𝐴⟶On |
9 | fvresi 5545 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
10 | 9 | adantr 272 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥) |
11 | fvresi 5545 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦) | |
12 | 11 | adantl 273 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦) |
13 | 10, 12 | eleq12d 2170 | . . 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 4810 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
16 | 8, 3, 14, 15 | issmo 6115 | 1 ⊢ Smo ( I ↾ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1299 ∈ wcel 1448 ⊆ wss 3021 I cid 4148 Ord word 4222 Oncon0 4223 ran crn 4478 ↾ cres 4479 Fn wfn 5054 ⟶wf 5055 ‘cfv 5059 Smo wsmo 6112 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-smo 6113 |
This theorem is referenced by: smo0 6125 |
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