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Mirrors > Home > MPE Home > Th. List > smofvon2 | Structured version Visualization version GIF version |
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Ref | Expression |
---|---|
smofvon2 | ⊢ (Smo 𝐹 → (𝐹‘𝐵) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsmo2 8386 | . . . 4 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) | |
2 | 1 | simp1bi 1144 | . . 3 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On) |
3 | ffvelcdm 7101 | . . . 4 ⊢ ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹‘𝐵) ∈ On) | |
4 | 3 | expcom 413 | . . 3 ⊢ (𝐵 ∈ dom 𝐹 → (𝐹:dom 𝐹⟶On → (𝐹‘𝐵) ∈ On)) |
5 | 2, 4 | syl5 34 | . 2 ⊢ (𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹‘𝐵) ∈ On)) |
6 | ndmfv 6942 | . . . 4 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅) | |
7 | 0elon 6440 | . . . 4 ⊢ ∅ ∈ On | |
8 | 6, 7 | eqeltrdi 2847 | . . 3 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) ∈ On) |
9 | 8 | a1d 25 | . 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹‘𝐵) ∈ On)) |
10 | 5, 9 | pm2.61i 182 | 1 ⊢ (Smo 𝐹 → (𝐹‘𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 ∀wral 3059 ∅c0 4339 dom cdm 5689 Ord word 6385 Oncon0 6386 ⟶wf 6559 ‘cfv 6563 Smo wsmo 8384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-smo 8385 |
This theorem is referenced by: smo11 8403 smoord 8404 smoword 8405 smogt 8406 |
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