Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dnwech Structured version   Visualization version   GIF version

Theorem dnwech 40873
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 40872 . . . 4 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
5 f1f1orn 6727 . . . 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 6670 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
8 frn 6607 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
94, 7, 83syl 18 . . . 4 (𝜑 → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
10 epweon 7625 . . . 4 E We On
11 wess 5576 . . . 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 2738 . . . 4 {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} = {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)}
1413f1owe 7224 . . 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 6787 . . . . . . . . 9 ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ∈ V
1716epeli 5497 . . . . . . . 8 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))
181, 2, 3dnnumch3lem 40871 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1918adantrr 714 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
201, 2, 3dnnumch3lem 40871 . . . . . . . . . 10 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2120adantrl 713 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2219, 21eleq12d 2833 . . . . . . . 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 5140 . . . . 5 (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))} = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))})
26 incom 4135 . . . . . 6 (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27 df-xp 5595 . . . . . . 7 (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)}
28 dnwech.h . . . . . . 7 𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}
2927, 28ineq12i 4144 . . . . . 6 ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})})
30 inopab 5739 . . . . . 6 ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
3126, 29, 303eqtri 2770 . . . . 5 (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
32 incom 4135 . . . . . 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 4142 . . . . . 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 5739 . . . . . 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 2770 . . . . 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 2803 . . . 4 (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37 weeq1 5577 . . . 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 5671 . . 3 (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40 weinxp 5671 . . 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 256 1 (𝜑𝐻 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  Vcvv 3432  cdif 3884  cin 3886  wss 3887  c0 4256  𝒫 cpw 4533  {csn 4561   cint 4879   class class class wbr 5074  {copab 5136  cmpt 5157   E cep 5494   We wwe 5543   × cxp 5587  ccnv 5588  ran crn 5590  cima 5592  Oncon0 6266  wf 6429  1-1wf1 6430  1-1-ontowf1o 6432  cfv 6433  recscrecs 8201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202
This theorem is referenced by:  aomclem3  40881
  Copyright terms: Public domain W3C validator