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Theorem fnwe2 39531
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 7815 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 711 . . . . . 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 767 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎𝐴)
10 simprr 769 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
111, 2, 4, 6, 8, 9, 10fnwe2lem2 39529 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑎 ≠ ∅)) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐)
1211ex 413 . . . 4 (𝜑 → ((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1312alrimiv 1919 . . 3 (𝜑 → ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
14 df-fr 5507 . . 3 (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑐𝑎𝑑𝑎 ¬ 𝑑𝑇𝑐))
1513, 14sylibr 235 . 2 (𝜑𝑇 Fr 𝐴)
163adantlr 711 . . . 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 767 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑎𝐴)
20 simprr 769 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝐴)
211, 2, 16, 17, 18, 19, 20fnwe2lem3 39530 . . 3 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
2221ralrimivva 3188 . 2 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
23 dfwe2 7485 . 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 396  wo 841  w3o 1078  wal 1526   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  wss 3933  c0 4288   class class class wbr 5057  {copab 5119   Fr wfr 5504   We wwe 5506  cres 5550  wf 6344  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  aomclem4  39535
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