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Theorem dnwech 43060
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 (𝜑𝐴𝑉)
dnnumch.g (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
dnwech.h 𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}
Assertion
Ref Expression
dnwech (𝜑𝐻 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.
StepHypRef Expression
1 dnnumch.f . . . . 5 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2 dnnumch.a . . . . 5 (𝜑𝐴𝑉)
3 dnnumch.g . . . . 5 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
41, 2, 3dnnumch3 43059 . . . 4 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
5 f1f1orn 6859 . . . 4 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1-onto→ran (𝑥𝐴 (𝐹 “ {𝑥})))
64, 5syl 17 . . 3 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1-onto→ran (𝑥𝐴 (𝐹 “ {𝑥})))
7 f1f 6804 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
8 frn 6743 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
94, 7, 83syl 18 . . . 4 (𝜑 → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
10 epweon 7795 . . . 4 E We On
11 wess 5671 . . . 4 (ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On → ( E We On → E We ran (𝑥𝐴 (𝐹 “ {𝑥}))))
129, 10, 11mpisyl 21 . . 3 (𝜑 → E We ran (𝑥𝐴 (𝐹 “ {𝑥})))
13 eqid 2737 . . . 4 {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} = {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)}
1413f1owe 7373 . . 3 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1-onto→ran (𝑥𝐴 (𝐹 “ {𝑥})) → ( E We ran (𝑥𝐴 (𝐹 “ {𝑥})) → {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
156, 12, 14sylc 65 . 2 (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} We 𝐴)
16 fvex 6919 . . . . . . . . 9 ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ∈ V
1716epeli 5586 . . . . . . . 8 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))
181, 2, 3dnnumch3lem 43058 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1918adantrr 717 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
201, 2, 3dnnumch3lem 43058 . . . . . . . . . 10 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2120adantrl 716 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2219, 21eleq12d 2835 . . . . . . . 8 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})))
2317, 22bitr2id 284 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)))
2423pm5.32da 579 . . . . . 6 (𝜑 → (((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})) ↔ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))))
2524opabbidv 5209 . . . . 5 (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))} = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))})
26 incom 4209 . . . . . 6 (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27 df-xp 5691 . . . . . . 7 (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)}
28 dnwech.h . . . . . . 7 𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}
2927, 28ineq12i 4218 . . . . . 6 ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})})
30 inopab 5839 . . . . . 6 ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
3126, 29, 303eqtri 2769 . . . . 5 (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
32 incom 4209 . . . . . 6 ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)})
3327ineq1i 4216 . . . . . 6 ((𝐴 × 𝐴) ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)}) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)})
34 inopab 5839 . . . . . 6 ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))}
3532, 33, 343eqtri 2769 . . . . 5 ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))}
3625, 31, 353eqtr4g 2802 . . . 4 (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37 weeq1 5672 . . . 4 ((𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) → ((𝐻 ∩ (𝐴 × 𝐴)) We 𝐴 ↔ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
3836, 37syl 17 . . 3 (𝜑 → ((𝐻 ∩ (𝐴 × 𝐴)) We 𝐴 ↔ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴))
39 weinxp 5770 . . 3 (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40 weinxp 5770 . . 3 ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} We 𝐴 ↔ ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)
4138, 39, 403bitr4g 314 . 2 (𝜑 → (𝐻 We 𝐴 ↔ {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
4215, 41mpbird 257 1 (𝜑𝐻 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cint 4946   class class class wbr 5143  {copab 5205  cmpt 5225   E cep 5583   We wwe 5636   × cxp 5683  ccnv 5684  ran crn 5686  cima 5688  Oncon0 6384  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  recscrecs 8410
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411
This theorem is referenced by:  aomclem3  43068
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