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Theorem dnwech 38120
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 38119 . . . 4 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
5 f1f1orn 6367 . . . 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 6319 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
8 frn 6265 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
94, 7, 83syl 18 . . . 4 (𝜑 → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
10 epweon 7216 . . . 4 E We On
11 wess 5305 . . . 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 2813 . . . 4 {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} = {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)}
1413f1owe 6830 . . 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 6424 . . . . . . . . 9 ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ∈ V
1716epeli 5233 . . . . . . . 8 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))
181, 2, 3dnnumch3lem 38118 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1918adantrr 699 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
201, 2, 3dnnumch3lem 38118 . . . . . . . . . 10 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2120adantrl 698 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2219, 21eleq12d 2886 . . . . . . . 8 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})))
2317, 22syl5rbb 275 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)))
2423pm5.32da 570 . . . . . 6 (𝜑 → (((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})) ↔ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))))
2524opabbidv 4917 . . . . 5 (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))} = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))})
26 incom 4011 . . . . . 6 (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27 df-xp 5324 . . . . . . 7 (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)}
28 dnwech.h . . . . . . 7 𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}
2927, 28ineq12i 4018 . . . . . 6 ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})})
30 inopab 5461 . . . . . 6 ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
3126, 29, 303eqtri 2839 . . . . 5 (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
32 incom 4011 . . . . . 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 4016 . . . . . 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 5461 . . . . . 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 2839 . . . . 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 2872 . . . 4 (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37 weeq1 5306 . . . 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 5395 . . 3 (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40 weinxp 5395 . . 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 305 . 2 (𝜑 → (𝐻 We 𝐴 ↔ {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
4215, 41mpbird 248 1 (𝜑𝐻 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wne 2985  wral 3103  Vcvv 3398  cdif 3773  cin 3775  wss 3776  c0 4123  𝒫 cpw 4358  {csn 4377   cint 4676   class class class wbr 4851  {copab 4913  cmpt 4930   E cep 5230   We wwe 5276   × cxp 5316  ccnv 5317  ran crn 5319  cima 5321  Oncon0 5943  wf 6100  1-1wf1 6101  1-1-ontowf1o 6103  cfv 6104  recscrecs 7706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-wrecs 7645  df-recs 7707
This theorem is referenced by:  aomclem3  38128
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