Theorem fnwe2 42250

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 8112 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 = (𝐹‘𝑥) → 𝑆 = 𝑈)
fnwe2.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}
fnwe2.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
fnwe2.f	⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
fnwe2.r	⊢ (𝜑 → 𝑅 We 𝐵)

Assertion

Ref	Expression
fnwe2	⊢ (𝜑 → 𝑇 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.

Step	Hyp	Ref	Expression
1		fnwe2.su	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
2		fnwe2.t	. . . . . 6 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}
3		fnwe2.s	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
4	3	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
5		fnwe2.f	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
7		fnwe2.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝐵)
8	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑅 We 𝐵)
9		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ⊆ 𝐴)
10		simprr 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
11	1, 2, 4, 6, 8, 9, 10	fnwe2lem2 42248	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)
12	11	ex 412	. . . 4 ⊢ (𝜑 → ((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
13	12	alrimiv 1922	. . 3 ⊢ (𝜑 → ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
14		df-fr 5621	. . 3 ⊢ (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
15	13, 14	sylibr 233	. 2 ⊢ (𝜑 → 𝑇 Fr 𝐴)
16	3	adantlr 712	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
17	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
18	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑅 We 𝐵)
19		simprl 768	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑎 ∈ 𝐴)
20		simprr 770	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏 ∈ 𝐴)
21	1, 2, 16, 17, 18, 19, 20	fnwe2lem3 42249	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
22	21	ralrimivva 3192	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
23		dfwe2 7754	. 2 ⊢ (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎)))
24	15, 22, 23	sylanbrc 582	1 ⊢ (𝜑 → 𝑇 We 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∨ w3o 1083  ∀wal 1531   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062  {crab 3424   ⊆ wss 3940  ∅c0 4314   class class class wbr 5138  {copab 5200   Fr wfr 5618   We wwe 5620   ↾ cres 5668  ⟶wf 6529  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541

This theorem is referenced by:  aomclem4  42254