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Theorem fnwe2 43042
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 8156 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
fnwe2.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
fnwe2.s ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
fnwe2.f (𝜑 → (𝐹𝐴):𝐴𝐵)
fnwe2.r (𝜑𝑅 We 𝐵)
Assertion
Ref Expression
fnwe2 (𝜑𝑇 We 𝐴)
Distinct variable groups:   𝑦,𝑈,𝑧   𝑥,𝑆,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧)   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnwe2.su . . . . . 6 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
2 fnwe2.t . . . . . 6 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
3 fnwe2.s . . . . . . 7 ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
43adantlr 715 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) ∧ 𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
5 fnwe2.f . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴𝐵)
65adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → (𝐹𝐴):𝐴𝐵)
7 fnwe2.r . . . . . . 7 (𝜑𝑅 We 𝐵)
87adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑅 We 𝐵)
9 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎𝐴)
10 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
111, 2, 4, 6, 8, 9, 10fnwe2lem2 43040 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐)
1211ex 412 . . . 4 (𝜑 → ((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1312alrimiv 1925 . . 3 (𝜑 → ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
14 df-fr 5641 . . 3 (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1513, 14sylibr 234 . 2 (𝜑𝑇 Fr 𝐴)
163adantlr 715 . . . 4 (((𝜑 ∧ (𝑎𝐴𝑏𝐴)) ∧ 𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
175adantr 480 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝐹𝐴):𝐴𝐵)
187adantr 480 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑅 We 𝐵)
19 simprl 771 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
20 simprr 773 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
211, 2, 16, 17, 18, 19, 20fnwe2lem3 43041 . . 3 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
2221ralrimivva 3200 . 2 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
23 dfwe2 7793 . 2 (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎)))
2415, 22, 23sylanbrc 583 1 (𝜑𝑇 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3o 1085  wal 1535   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  wss 3963  c0 4339   class class class wbr 5148  {copab 5210   Fr wfr 5638   We wwe 5640  cres 5691  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  aomclem4  43046
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