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Mirrors > Home > MPE Home > Th. List > smodm2 | Structured version Visualization version 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 7977 | . 2 ⊢ (Smo 𝐹 → Ord dom 𝐹) | |
2 | fndm 6448 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | ordeq 6191 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (Ord dom 𝐹 ↔ Ord 𝐴)) |
5 | 4 | biimpa 477 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴) |
6 | 1, 5 | sylan2 592 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 dom cdm 5548 Ord word 6183 Fn wfn 6343 Smo wsmo 7971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ral 3140 df-rex 3141 df-in 3940 df-ss 3949 df-uni 4831 df-tr 5164 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-fn 6351 df-smo 7972 |
This theorem is referenced by: smo11 7990 smoord 7991 smoword 7992 smogt 7993 smorndom 7994 coftr 9683 |
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