Theorem dnwech 43044

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 43043	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)
5		f1f1orn 6814	. . . 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 6759	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On)
8		frn 6698	. . . . 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 7754	. . . 4 ⊢ E We On
11		wess 5627	. . . 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 2730	. . . 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 7331	. . 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 6874	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ∈ V
17	16	epeli 5543	. . . . . . . 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 43042	. . . . . . . . . 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 43042	. . . . . . . . . 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 2823	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})))
23	17, 22	bitr2id 284	. . . . . . 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 579	. . . . . 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 5176	. . . . 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 4175	. . . . . 6 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27		df-xp 5647	. . . . . . 7 ⊢ (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)}
28		dnwech.h	. . . . . . 7 ⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}
29	27, 28	ineq12i 4184	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})})
30		inopab 5795	. . . . . 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 2757	. . . . 5 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤}))}
32		incom 4175	. . . . . 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 4182	. . . . . 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 5795	. . . . . 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 2757	. . . . 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 2790	. . . 4 ⊢ (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37		weeq1 5628	. . . 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 5726	. . 3 ⊢ (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40		weinxp 5726	. . 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 314	. 2 ⊢ (𝜑 → (𝐻 We 𝐴 ↔ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
42	15, 41	mpbird 257	1 ⊢ (𝜑 → 𝐻 We 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  Vcvv 3450   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ∩ cint 4913   class class class wbr 5110  {copab 5172   ↦ cmpt 5191   E cep 5540   We wwe 5593   × cxp 5639  ^◡ccnv 5640  ran crn 5642   “ cima 5644  Oncon0 6335  ⟶wf 6510  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514  recscrecs 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343

This theorem is referenced by:  aomclem3  43052