Mathbox for Stefan O'Rear |
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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 7828 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 713 | . . . . . 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 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵) |
7 | fnwe2.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝐵) | |
8 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑅 We 𝐵) |
9 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ⊆ 𝐴) | |
10 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ≠ ∅) | |
11 | 1, 2, 4, 6, 8, 9, 10 | fnwe2lem2 39658 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐) |
12 | 11 | ex 415 | . . . 4 ⊢ (𝜑 → ((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)) |
13 | 12 | alrimiv 1928 | . . 3 ⊢ (𝜑 → ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)) |
14 | df-fr 5516 | . . 3 ⊢ (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)) | |
15 | 13, 14 | sylibr 236 | . 2 ⊢ (𝜑 → 𝑇 Fr 𝐴) |
16 | 3 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) |
17 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵) |
18 | 7 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑅 We 𝐵) |
19 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑎 ∈ 𝐴) | |
20 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏 ∈ 𝐴) | |
21 | 1, 2, 16, 17, 18, 19, 20 | fnwe2lem3 39659 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎)) |
22 | 21 | ralrimivva 3193 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎)) |
23 | dfwe2 7498 | . 2 ⊢ (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))) | |
24 | 15, 22, 23 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝑇 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 {copab 5130 Fr wfr 5513 We wwe 5515 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 |
This theorem is referenced by: aomclem4 39664 |
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