Theorem dnwech 43630

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 43629	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)
5		f1f1orn 6820	. . . 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 6762	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On)
8		frn 6701	. . . . 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 7760	. . . 4 ⊢ E We On
11		wess 5635	. . . 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 2764	. . . 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 7339	. . 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 6882	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ∈ V
17	16	epeli 5551	. . . . . . . 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 43628	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
19	18	adantrr 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
20	1, 2, 3	dnnumch3lem 43628	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
21	20	adantrl 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
22	19, 21	eleq12d 2858	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})))
23	17, 22	bitr2id 286	. . . . . . 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 587	. . . . . 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 5168	. . . . 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 4163	. . . . . 6 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27		df-xp 5655	. . . . . . 7 ⊢ (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)}
28		dnwech.h	. . . . . . 7 ⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}
29	27, 28	ineq12i 4172	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})})
30		inopab 5804	. . . . . 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 2791	. . . . 5 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤}))}
32		incom 4163	. . . . . 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 4170	. . . . . 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 5804	. . . . . 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 2791	. . . . 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 2824	. . . 4 ⊢ (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37		weeq1 5636	. . . 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 5734	. . 3 ⊢ (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40		weinxp 5734	. . 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 316	. 2 ⊢ (𝜑 → (𝐻 We 𝐴 ↔ {⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
42	15, 41	mpbird 259	1 ⊢ (𝜑 → 𝐻 We 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  Vcvv 3456   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  {csn 4584  ∩ cint 4907   class class class wbr 5102  {copab 5164   ↦ cmpt 5183   E cep 5548   We wwe 5601   × cxp 5647  ^◡ccnv 5648  ran crn 5650   “ cima 5652  Oncon0 6348  ⟶wf 6519  –1-1→wf1 6520  –1-1-onto→wf1o 6522  ‘cfv 6523  recscrecs 8343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-ov 7401  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344

This theorem is referenced by:  aomclem3  43638