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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 7331 | . . . 4 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) | |
2 | 1 | simp1bi 1069 | . . 3 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On) |
3 | ffvelrn 6265 | . . . 4 ⊢ ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹‘𝐵) ∈ On) | |
4 | 3 | expcom 450 | . . 3 ⊢ (𝐵 ∈ dom 𝐹 → (𝐹:dom 𝐹⟶On → (𝐹‘𝐵) ∈ On)) |
5 | 2, 4 | syl5 33 | . 2 ⊢ (𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹‘𝐵) ∈ On)) |
6 | ndmfv 6128 | . . . 4 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅) | |
7 | 0elon 5695 | . . . 4 ⊢ ∅ ∈ On | |
8 | 6, 7 | syl6eqel 2696 | . . 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 175 | 1 ⊢ (Smo 𝐹 → (𝐹‘𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 dom cdm 5038 Ord word 5639 Oncon0 5640 ⟶wf 5800 ‘cfv 5804 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-smo 7330 |
This theorem is referenced by: smo11 7348 smoord 7349 smoword 7350 smogt 7351 |
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