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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 8353 | . . . 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 7083 | . . . 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 6926 | . . . 4 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅) | |
7 | 0elon 6418 | . . . 4 ⊢ ∅ ∈ On | |
8 | 6, 7 | eqeltrdi 2840 | . . 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 2105 ∀wral 3060 ∅c0 4322 dom cdm 5676 Ord word 6363 Oncon0 6364 ⟶wf 6539 ‘cfv 6543 Smo wsmo 8351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-id 5574 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-ord 6367 df-on 6368 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-smo 8352 |
This theorem is referenced by: smo11 8370 smoord 8371 smoword 8372 smogt 8373 |
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