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Mirrors > Home > ILE Home > Th. List > smodm2 | GIF version |
Description: The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Ref | Expression |
---|---|
smodm2 | ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smodm 6188 | . 2 ⊢ (Smo 𝐹 → Ord dom 𝐹) | |
2 | fndm 5222 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | ordeq 4294 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴)) |
5 | 4 | biimpa 294 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴) |
6 | 1, 5 | sylan2 284 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 Ord word 4284 dom cdm 4539 Fn wfn 5118 Smo wsmo 6182 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-in 3077 df-ss 3084 df-uni 3737 df-tr 4027 df-iord 4288 df-fn 5126 df-smo 6183 |
This theorem is referenced by: (None) |
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