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Mirrors > Home > MPE Home > Th. List > iordsmo | Structured version Visualization version 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 6448 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
2 | rnresi 5910 | . . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴 | |
3 | iordsmo.1 | . . . . 5 ⊢ Ord 𝐴 | |
4 | ordsson 7484 | . . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐴 ⊆ On |
6 | 2, 5 | eqsstri 3949 | . . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On |
7 | df-f 6328 | . . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On)) | |
8 | 1, 6, 7 | mpbir2an 710 | . 2 ⊢ ( I ↾ 𝐴):𝐴⟶On |
9 | fvresi 6912 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥) |
11 | fvresi 6912 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦) | |
12 | 11 | adantl 485 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦) |
13 | 10, 12 | eleq12d 2884 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦)) |
14 | 13 | biimprd 251 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦))) |
15 | dmresi 5888 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
16 | 8, 3, 14, 15 | issmo 7968 | 1 ⊢ Smo ( I ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 I cid 5424 ran crn 5520 ↾ cres 5521 Ord word 6158 Oncon0 6159 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 Smo wsmo 7965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-smo 7966 |
This theorem is referenced by: smo0 7978 |
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