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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 6697 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
| 2 | rnresi 6093 | . . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴 | |
| 3 | iordsmo.1 | . . . . 5 ⊢ Ord 𝐴 | |
| 4 | ordsson 7803 | . . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
| 5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐴 ⊆ On | 
| 6 | 2, 5 | eqsstri 4030 | . . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On | 
| 7 | df-f 6565 | . . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On)) | |
| 8 | 1, 6, 7 | mpbir2an 711 | . 2 ⊢ ( I ↾ 𝐴):𝐴⟶On | 
| 9 | fvresi 7193 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
| 10 | 9 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥) | 
| 11 | fvresi 7193 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦) | |
| 12 | 11 | adantl 481 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦) | 
| 13 | 10, 12 | eleq12d 2835 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦)) | 
| 14 | 13 | biimprd 248 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦))) | 
| 15 | dmresi 6070 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
| 16 | 8, 3, 14, 15 | issmo 8388 | 1 ⊢ Smo ( I ↾ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 I cid 5577 ran crn 5686 ↾ cres 5687 Ord word 6383 Oncon0 6384 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Smo wsmo 8385 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-smo 8386 | 
| This theorem is referenced by: smo0 8398 | 
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