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Mirrors > Home > MPE Home > Th. List > smocdmdom | Structured version Visualization version GIF version |
Description: The codomain of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.) |
Ref | Expression |
---|---|
smocdmdom | ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1191 | . . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffnd 6705 | . . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴) |
3 | simpl2 1192 | . . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Smo 𝐹) | |
4 | smodm2 8337 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) | |
5 | 2, 3, 4 | syl2anc 584 | . . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Ord 𝐴) |
6 | ordelord 6375 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) | |
7 | 5, 6 | sylancom 588 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) |
8 | simpl3 1193 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Ord 𝐵) | |
9 | simpr 485 | . . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
10 | smogt 8349 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥)) | |
11 | 2, 3, 9, 10 | syl3anc 1371 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥)) |
12 | ffvelcdm 7068 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
13 | 12 | 3ad2antl1 1185 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
14 | ordtr2 6397 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝐵) → ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝑥 ∈ 𝐵)) | |
15 | 14 | imp 407 | . . . 4 ⊢ (((Ord 𝑥 ∧ Ord 𝐵) ∧ (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝐵)) → 𝑥 ∈ 𝐵) |
16 | 7, 8, 11, 13, 15 | syl22anc 837 | . . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
17 | 16 | ex 413 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
18 | 17 | ssrdv 3984 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ⊆ wss 3944 Ord word 6352 Fn wfn 6527 ⟶wf 6528 ‘cfv 6532 Smo wsmo 8327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-ord 6356 df-on 6357 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-smo 8328 |
This theorem is referenced by: cofsmo 10246 hsmexlem1 10403 |
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