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Mirrors > Home > ILE Home > Th. List > smores3 | GIF version |
Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
smores3 | ⊢ ((Smo (𝐴 ↾ 𝐵) ∧ 𝐶 ∈ (dom 𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Smo (𝐴 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 4946 | . . . . . 6 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
2 | incom 3342 | . . . . . 6 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵) | |
3 | 1, 2 | eqtri 2210 | . . . . 5 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵) |
4 | 3 | eleq2i 2256 | . . . 4 ⊢ (𝐶 ∈ dom (𝐴 ↾ 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∩ 𝐵)) |
5 | smores 6318 | . . . 4 ⊢ ((Smo (𝐴 ↾ 𝐵) ∧ 𝐶 ∈ dom (𝐴 ↾ 𝐵)) → Smo ((𝐴 ↾ 𝐵) ↾ 𝐶)) | |
6 | 4, 5 | sylan2br 288 | . . 3 ⊢ ((Smo (𝐴 ↾ 𝐵) ∧ 𝐶 ∈ (dom 𝐴 ∩ 𝐵)) → Smo ((𝐴 ↾ 𝐵) ↾ 𝐶)) |
7 | 6 | 3adant3 1019 | . 2 ⊢ ((Smo (𝐴 ↾ 𝐵) ∧ 𝐶 ∈ (dom 𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Smo ((𝐴 ↾ 𝐵) ↾ 𝐶)) |
8 | inss2 3371 | . . . . . 6 ⊢ (dom 𝐴 ∩ 𝐵) ⊆ 𝐵 | |
9 | 8 | sseli 3166 | . . . . 5 ⊢ (𝐶 ∈ (dom 𝐴 ∩ 𝐵) → 𝐶 ∈ 𝐵) |
10 | ordelss 4397 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐶 ⊆ 𝐵) | |
11 | 10 | ancoms 268 | . . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ Ord 𝐵) → 𝐶 ⊆ 𝐵) |
12 | 9, 11 | sylan 283 | . . . 4 ⊢ ((𝐶 ∈ (dom 𝐴 ∩ 𝐵) ∧ Ord 𝐵) → 𝐶 ⊆ 𝐵) |
13 | 12 | 3adant1 1017 | . . 3 ⊢ ((Smo (𝐴 ↾ 𝐵) ∧ 𝐶 ∈ (dom 𝐴 ∩ 𝐵) ∧ Ord 𝐵) → 𝐶 ⊆ 𝐵) |
14 | resabs1 4954 | . . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐶)) | |
15 | smoeq 6316 | . . 3 ⊢ (((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐶) → (Smo ((𝐴 ↾ 𝐵) ↾ 𝐶) ↔ Smo (𝐴 ↾ 𝐶))) | |
16 | 13, 14, 15 | 3syl 17 | . 2 ⊢ ((Smo (𝐴 ↾ 𝐵) ∧ 𝐶 ∈ (dom 𝐴 ∩ 𝐵) ∧ Ord 𝐵) → (Smo ((𝐴 ↾ 𝐵) ↾ 𝐶) ↔ Smo (𝐴 ↾ 𝐶))) |
17 | 7, 16 | mpbid 147 | 1 ⊢ ((Smo (𝐴 ↾ 𝐵) ∧ 𝐶 ∈ (dom 𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Smo (𝐴 ↾ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∩ cin 3143 ⊆ wss 3144 Ord word 4380 dom cdm 4644 ↾ cres 4646 Smo wsmo 6311 |
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 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-tr 4117 df-iord 4384 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-smo 6312 |
This theorem is referenced by: (None) |
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