Theorem fnwe2 43010

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 8173 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 714	. . . . . 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 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ⊆ 𝐴)
10		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)) → 𝑎 ≠ ∅)
11	1, 2, 4, 6, 8, 9, 10	fnwe2lem2 43008	. . . . 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 1926	. . 3 ⊢ (𝜑 → ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
14		df-fr 5652	. . 3 ⊢ (𝑇 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑎 ¬ 𝑑𝑇𝑐))
15	13, 14	sylibr 234	. 2 ⊢ (𝜑 → 𝑇 Fr 𝐴)
16	3	adantlr 714	. . . 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 770	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑎 ∈ 𝐴)
20		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏 ∈ 𝐴)
21	1, 2, 16, 17, 18, 19, 20	fnwe2lem3 43009	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
22	21	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
23		dfwe2 7809	. 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 846   ∨ w3o 1086  ∀wal 1535   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166  {copab 5228   Fr wfr 5649   We wwe 5651   ↾ cres 5702  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  aomclem4  43014