Theorem fnwe2 43840

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 8134 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 728	. . . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
7		fnwe2.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝐵)
8	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑅 We 𝐵)
9		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ⊆ 𝐴)
10		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
11	1, 2, 4, 6, 8, 9, 10	fnwe2lem2 43838	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐)
12	11	ex 418	. . . 4 ⊢ (𝜑 → ((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
13	12	alrimiv 1960	. . 3 ⊢ (𝜑 → ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
14		df-fr 5616	. . 3 ⊢ (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
15	13, 14	sylibr 237	. 2 ⊢ (𝜑 → 𝑇 Fr 𝐴)
16	3	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
17	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝐹 ↾ 𝐴):𝐴⟶𝐵)
18	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑅 We 𝐵)
19		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑎 ∈ 𝐴)
20		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏 ∈ 𝐴)
21	1, 2, 16, 17, 18, 19, 20	fnwe2lem3 43839	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
22	21	ralrimivva 3210	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
23		dfwe2 7779	. 2 ⊢ (𝑇 We 𝐴 ↔ (𝑇 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎)))
24	15, 22, 23	sylanbrc 595	1 ⊢ (𝜑 → 𝑇 We 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102  ∀wal 1568   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  {copab 5175   Fr wfr 5613   We wwe 5615   ↾ cres 5665  ⟶wf 6536  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548

This theorem is used by:  aomclem4  43844