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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 7335 | . 2 ⊢ (Smo 𝐹 → Ord dom 𝐹) | |
2 | fndm 5904 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | ordeq 5647 | . . . 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 500 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Ord dom 𝐹) → Ord 𝐴) |
6 | 1, 5 | sylan2 490 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 dom cdm 5038 Ord word 5639 Fn wfn 5799 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-fn 5807 df-smo 7330 |
This theorem is referenced by: smo11 7348 smoord 7349 smoword 7350 smogt 7351 smorndom 7352 coftr 8978 |
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