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Mirrors > Home > ILE Home > Th. List > issmo | GIF version |
Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Ref | Expression |
---|---|
issmo.1 | ⊢ 𝐴:𝐵⟶On |
issmo.2 | ⊢ Ord 𝐵 |
issmo.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) |
issmo.4 | ⊢ dom 𝐴 = 𝐵 |
Ref | Expression |
---|---|
issmo | ⊢ Smo 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmo.1 | . . 3 ⊢ 𝐴:𝐵⟶On | |
2 | issmo.4 | . . . 4 ⊢ dom 𝐴 = 𝐵 | |
3 | 2 | feq2i 5274 | . . 3 ⊢ (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On) |
4 | 1, 3 | mpbir 145 | . 2 ⊢ 𝐴:dom 𝐴⟶On |
5 | issmo.2 | . . 3 ⊢ Ord 𝐵 | |
6 | ordeq 4302 | . . . 4 ⊢ (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵)) | |
7 | 2, 6 | ax-mp 5 | . . 3 ⊢ (Ord dom 𝐴 ↔ Ord 𝐵) |
8 | 5, 7 | mpbir 145 | . 2 ⊢ Ord dom 𝐴 |
9 | 2 | eleq2i 2207 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵) |
10 | 2 | eleq2i 2207 | . . . 4 ⊢ (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ 𝐵) |
11 | issmo.3 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) | |
12 | 9, 10, 11 | syl2anb 289 | . . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) |
13 | 12 | rgen2a 2489 | . 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) |
14 | df-smo 6191 | . 2 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))) | |
15 | 4, 8, 13, 14 | mpbir3an 1164 | 1 ⊢ Smo 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∀wral 2417 Ord word 4292 Oncon0 4293 dom cdm 4547 ⟶wf 5127 ‘cfv 5131 Smo wsmo 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-in 3082 df-ss 3089 df-uni 3745 df-tr 4035 df-iord 4296 df-fn 5134 df-f 5135 df-smo 6191 |
This theorem is referenced by: iordsmo 6202 |
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