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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 5922 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
2 | rnresi 5398 | . . . 4 ⊢ ran ( I ↾ 𝐴) = 𝐴 | |
3 | iordsmo.1 | . . . . 5 ⊢ Ord 𝐴 | |
4 | ordsson 6881 | . . . . 5 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐴 ⊆ On |
6 | 2, 5 | eqsstri 3598 | . . 3 ⊢ ran ( I ↾ 𝐴) ⊆ On |
7 | df-f 5808 | . . 3 ⊢ (( I ↾ 𝐴):𝐴⟶On ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ran ( I ↾ 𝐴) ⊆ On)) | |
8 | 1, 6, 7 | mpbir2an 957 | . 2 ⊢ ( I ↾ 𝐴):𝐴⟶On |
9 | fvresi 6344 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑥) = 𝑥) |
11 | fvresi 6344 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦) | |
12 | 11 | adantl 481 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (( I ↾ 𝐴)‘𝑦) = 𝑦) |
13 | 10, 12 | eleq12d 2682 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦) ↔ 𝑥 ∈ 𝑦)) |
14 | 13 | biimprd 237 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (( I ↾ 𝐴)‘𝑥) ∈ (( I ↾ 𝐴)‘𝑦))) |
15 | dmresi 5376 | . 2 ⊢ dom ( I ↾ 𝐴) = 𝐴 | |
16 | 8, 3, 14, 15 | issmo 7332 | 1 ⊢ Smo ( I ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 I cid 4948 ran crn 5039 ↾ cres 5040 Ord word 5639 Oncon0 5640 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 Smo wsmo 7329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-smo 7330 |
This theorem is referenced by: smo0 7342 |
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