Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnwe2 Structured version   Visualization version   GIF version

Theorem fnwe2 41795
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 8118 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 714 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) ∧ 𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
5 fnwe2.f . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴𝐵)
65adantr 482 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → (𝐹𝐴):𝐴𝐵)
7 fnwe2.r . . . . . . 7 (𝜑𝑅 We 𝐵)
87adantr 482 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑅 We 𝐵)
9 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎𝐴)
10 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
111, 2, 4, 6, 8, 9, 10fnwe2lem2 41793 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐)
1211ex 414 . . . 4 (𝜑 → ((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1312alrimiv 1931 . . 3 (𝜑 → ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
14 df-fr 5632 . . 3 (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1513, 14sylibr 233 . 2 (𝜑𝑇 Fr 𝐴)
163adantlr 714 . . . 4 (((𝜑 ∧ (𝑎𝐴𝑏𝐴)) ∧ 𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
175adantr 482 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝐹𝐴):𝐴𝐵)
187adantr 482 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑅 We 𝐵)
19 simprl 770 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
20 simprr 772 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
211, 2, 16, 17, 18, 19, 20fnwe2lem3 41794 . . 3 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
2221ralrimivva 3201 . 2 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
23 dfwe2 7761 . 2 (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎)))
2415, 22, 23sylanbrc 584 1 (𝜑𝑇 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  w3o 1087  wal 1540   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  wss 3949  c0 4323   class class class wbr 5149  {copab 5211   Fr wfr 5629   We wwe 5631  cres 5679  wf 6540  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by:  aomclem4  41799
  Copyright terms: Public domain W3C validator