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Theorem fnwe2 37100
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 7238 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 750 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) ∧ 𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
5 fnwe2.f . . . . . . 7 (𝜑 → (𝐹𝐴):𝐴𝐵)
65adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → (𝐹𝐴):𝐴𝐵)
7 fnwe2.r . . . . . . 7 (𝜑𝑅 We 𝐵)
87adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑅 We 𝐵)
9 simprl 793 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎𝐴)
10 simprr 795 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
111, 2, 4, 6, 8, 9, 10fnwe2lem2 37098 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐)
1211ex 450 . . . 4 (𝜑 → ((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1312alrimiv 1852 . . 3 (𝜑 → ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
14 df-fr 5033 . . 3 (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1513, 14sylibr 224 . 2 (𝜑𝑇 Fr 𝐴)
163adantlr 750 . . . 4 (((𝜑 ∧ (𝑎𝐴𝑏𝐴)) ∧ 𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
175adantr 481 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝐹𝐴):𝐴𝐵)
187adantr 481 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑅 We 𝐵)
19 simprl 793 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
20 simprr 795 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
211, 2, 16, 17, 18, 19, 20fnwe2lem3 37099 . . 3 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
2221ralrimivva 2965 . 2 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
23 dfwe2 6928 . 2 (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎)))
2415, 22, 23sylanbrc 697 1 (𝜑𝑇 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3o 1035  wal 1478   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3555  c0 3891   class class class wbr 4613  {copab 4672   Fr wfr 5030   We wwe 5032  cres 5076  wf 5843  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855
This theorem is referenced by:  aomclem4  37104
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