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Theorem fnwe2 36641
 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 7180 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 747 . . . . . 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 790 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎𝐴)
10 simprr 792 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
111, 2, 4, 6, 8, 9, 10fnwe2lem2 36639 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐)
1211ex 449 . . . 4 (𝜑 → ((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1312alrimiv 1842 . . 3 (𝜑 → ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
14 df-fr 4997 . . 3 (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1513, 14sylibr 223 . 2 (𝜑𝑇 Fr 𝐴)
163adantlr 747 . . . 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 790 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
20 simprr 792 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
211, 2, 16, 17, 18, 19, 20fnwe2lem3 36640 . . 3 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
2221ralrimivva 2954 . 2 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
23 dfwe2 6873 . 2 (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎)))
2415, 22, 23sylanbrc 695 1 (𝜑𝑇 We 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  {copab 4642   Fr wfr 4994   We wwe 4996   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  aomclem4  36645
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