Theorem dnwech 39668

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 39667	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)
5		f1f1orn 6626	. . . 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 6520	. . . . 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 7497	. . . 4 ⊢ E We On
11		wess 5542	. . . 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 2821	. . . 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 7106	. . 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 6683	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ∈ V
17	16	epeli 5468	. . . . . . . 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 39666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
19	18	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
20	1, 2, 3	dnnumch3lem 39666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
21	20	adantrl 714	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
22	19, 21	eleq12d 2907	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})))
23	17, 22	syl5rbb 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 581	. . . . . 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 5132	. . . . 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 4178	. . . . . 6 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27		df-xp 5561	. . . . . . 7 ⊢ (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)}
28		dnwech.h	. . . . . . 7 ⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}
29	27, 28	ineq12i 4187	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})})
30		inopab 5701	. . . . . 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 2848	. . . . 5 ⊢ (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤}))}
32		incom 4178	. . . . . 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 4185	. . . . . 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 5701	. . . . . 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 2848	. . . . 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 2881	. . . 4 ⊢ (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) E ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37		weeq1 5543	. . . 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 5636	. . 3 ⊢ (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40		weinxp 5636	. . 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 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  ∩ cint 4876   class class class wbr 5066  {copab 5128   ↦ cmpt 5146   E cep 5464   We wwe 5513   × cxp 5553  ^◡ccnv 5554  ran crn 5556   “ cima 5558  Oncon0 6191  ⟶wf 6351  –1-1→wf1 6352  –1-1-onto→wf1o 6354  ‘cfv 6355  recscrecs 8007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-wrecs 7947  df-recs 8008

This theorem is referenced by:  aomclem3  39676