Theorem dnwech 40468

Description: Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
dnnumch.f	⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
dnnumch.g	⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
dnwech.h	⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}

Assertion

Ref	Expression
dnwech	⊢ (𝜑 → 𝐻 We 𝐴)

Distinct variable groups:   𝑣,𝐹,𝑤,𝑦   𝑣,𝐺,𝑤,𝑦,𝑧   𝑣,𝐴,𝑤,𝑦,𝑧   𝜑,𝑣,𝑤

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝐻(𝑦,𝑧,𝑤,𝑣)   𝑉(𝑦,𝑧,𝑤,𝑣)

Proof of Theorem dnwech

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnnumch.f	. . . . 5 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2		dnnumch.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		dnnumch.g	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
4	1, 2, 3	dnnumch3 40467	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)
5		f1f1orn 6632	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})))
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})))
7		f1f 6575	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On)
8		frn 6512	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On → ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) ⊆ On)
9	4, 7, 8	3syl 18	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) ⊆ On)
10		epweon 7519	. . . 4 ⊢ E We On
11		wess 5513	. . . 4 ⊢ (ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) ⊆ On → ( E We On → E We ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))))
12	9, 10, 11	mpisyl 21	. . 3 ⊢ (𝜑 → E We ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})))
13		eqid 2739	. . . 4 ⊢ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} = {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)}
14	13	f1owe 7122	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) → ( E We ran (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) → {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
15	6, 12, 14	sylc 65	. 2 ⊢ (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} We 𝐴)
16		fvex 6690	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ∈ V
17	16	epeli 5437	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤))
18	1, 2, 3	dnnumch3lem 40466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
19	18	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
20	1, 2, 3	dnnumch3lem 40466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
21	20	adantrl 716	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
22	19, 21	eleq12d 2828	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})))
23	17, 22	bitr2id 287	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤}) ↔ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)))
24	23	pm5.32da 582	. . . . . 6 ⊢ (𝜑 → (((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})) ↔ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤))))
25	24	opabbidv 5097	. . . . 5 ⊢ (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤}))} = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤))})
26		incom 4092	. . . . . 6 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27		df-xp 5532	. . . . . . 7 ⊢ (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)}
28		dnwech.h	. . . . . . 7 ⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}
29	27, 28	ineq12i 4102	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})})
30		inopab 5674	. . . . . 6 ⊢ ({⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤}))}
31	26, 29, 30	3eqtri 2766	. . . . 5 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤}))}
32		incom 4092	. . . . . 6 ⊢ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)})
33	27	ineq1i 4100	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)}) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)})
34		inopab 5674	. . . . . 6 ⊢ ({⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤))}
35	32, 33, 34	3eqtri 2766	. . . . 5 ⊢ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤))}
36	25, 31, 35	3eqtr4g 2799	. . . 4 ⊢ (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37		weeq1 5514	. . . 4 ⊢ ((𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) → ((𝐻 ∩ (𝐴 × 𝐴)) We 𝐴 ↔ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
38	36, 37	syl 17	. . 3 ⊢ (𝜑 → ((𝐻 ∩ (𝐴 × 𝐴)) We 𝐴 ↔ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
39		weinxp 5608	. . 3 ⊢ (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40		weinxp 5608	. . 3 ⊢ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} We 𝐴 ↔ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
41	38, 39, 40	3bitr4g 317	. 2 ⊢ (𝜑 → (𝐻 We 𝐴 ↔ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
42	15, 41	mpbird 260	1 ⊢ (𝜑 → 𝐻 We 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114   ≠ wne 2935  ∀wral 3054  Vcvv 3399   ∖ cdif 3841   ∩ cin 3843   ⊆ wss 3844  ∅c0 4212  𝒫 cpw 4489  {csn 4517  ∩ cint 4837   class class class wbr 5031  {copab 5093   ↦ cmpt 5111   E cep 5434   We wwe 5483   × cxp 5524  ^◡ccnv 5525  ran crn 5527   “ cima 5529  Oncon0 6173  ⟶wf 6336  –1-1→wf1 6337  –1-1-onto→wf1o 6339  ‘cfv 6340  recscrecs 8039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-wrecs 7979  df-recs 8040

This theorem is referenced by:  aomclem3  40476