![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5355 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
2 | rnresi 5006 | . . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴 | |
3 | iordsmo.1 | . . . . 5 ⊢ Ord 𝐴 | |
4 | ordsson 4512 | . . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐴 ⊆ On |
6 | 2, 5 | eqsstri 3202 | . . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On |
7 | df-f 5242 | . . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On)) | |
8 | 1, 6, 7 | mpbir2an 944 | . 2 ⊢ ( I ↾ 𝐴):𝐴⟶On |
9 | fvresi 5733 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
10 | 9 | adantr 276 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥) |
11 | fvresi 5733 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦) | |
12 | 11 | adantl 277 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦) |
13 | 10, 12 | eleq12d 2260 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦)) |
14 | 13 | biimprd 158 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦))) |
15 | dmresi 4983 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
16 | 8, 3, 14, 15 | issmo 6317 | 1 ⊢ Smo ( I ↾ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 I cid 4309 Ord word 4383 Oncon0 4384 ran crn 4648 ↾ cres 4649 Fn wfn 5233 ⟶wf 5234 ‘cfv 5238 Smo wsmo 6314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-tr 4120 df-id 4314 df-iord 4387 df-on 4389 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-fv 5246 df-smo 6315 |
This theorem is referenced by: smo0 6327 |
Copyright terms: Public domain | W3C validator |