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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 1188 | . . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffnd 6718 | . . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴) |
3 | simpl2 1189 | . . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Smo 𝐹) | |
4 | smodm2 8374 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) | |
5 | 2, 3, 4 | syl2anc 582 | . . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Ord 𝐴) |
6 | ordelord 6386 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) | |
7 | 5, 6 | sylancom 586 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) |
8 | simpl3 1190 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → Ord 𝐵) | |
9 | simpr 483 | . . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
10 | smogt 8386 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥)) | |
11 | 2, 3, 9, 10 | syl3anc 1368 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥)) |
12 | ffvelcdm 7086 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
13 | 12 | 3ad2antl1 1182 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
14 | ordtr2 6408 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝐵) → ((𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝑥 ∈ 𝐵)) | |
15 | 14 | imp 405 | . . . 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 411 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
18 | 17 | ssrdv 3978 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ⊆ wss 3939 Ord word 6363 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 Smo wsmo 8364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-ord 6367 df-on 6368 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-smo 8365 |
This theorem is referenced by: cofsmo 10292 hsmexlem1 10449 |
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