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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fnwe2 | Structured version Visualization version GIF version | ||
| Description: A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 8158 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| fnwe2.su | ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) | 
| fnwe2.t | ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} | 
| fnwe2.s | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) | 
| fnwe2.f | ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵) | 
| fnwe2.r | ⊢ (𝜑 → 𝑅 We 𝐵) | 
| Ref | Expression | 
|---|---|
| fnwe2 | ⊢ (𝜑 → 𝑇 We 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fnwe2.su | . . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) | |
| 2 | fnwe2.t | . . . . . 6 ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} | |
| 3 | fnwe2.s | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) | |
| 4 | 3 | adantlr 715 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) | 
| 5 | fnwe2.f | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵) | |
| 6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵) | 
| 7 | fnwe2.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝐵) | |
| 8 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑅 We 𝐵) | 
| 9 | simprl 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ⊆ 𝐴) | |
| 10 | simprr 772 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ≠ ∅) | |
| 11 | 1, 2, 4, 6, 8, 9, 10 | fnwe2lem2 43068 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐) | 
| 12 | 11 | ex 412 | . . . 4 ⊢ (𝜑 → ((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)) | 
| 13 | 12 | alrimiv 1926 | . . 3 ⊢ (𝜑 → ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)) | 
| 14 | df-fr 5636 | . . 3 ⊢ (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)) | |
| 15 | 13, 14 | sylibr 234 | . 2 ⊢ (𝜑 → 𝑇 Fr 𝐴) | 
| 16 | 3 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) | 
| 17 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵) | 
| 18 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑅 We 𝐵) | 
| 19 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑎 ∈ 𝐴) | |
| 20 | simprr 772 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏 ∈ 𝐴) | |
| 21 | 1, 2, 16, 17, 18, 19, 20 | fnwe2lem3 43069 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎)) | 
| 22 | 21 | ralrimivva 3201 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎)) | 
| 23 | dfwe2 7795 | . 2 ⊢ (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))) | |
| 24 | 15, 22, 23 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑇 We 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 ⊆ wss 3950 ∅c0 4332 class class class wbr 5142 {copab 5204 Fr wfr 5633 We wwe 5635 ↾ cres 5686 ⟶wf 6556 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 | 
| This theorem is referenced by: aomclem4 43074 | 
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