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Theorem dnwech 38144
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 38143 . . . 4 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
5 f1f1orn 6290 . . . 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 6242 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
8 frn 6190 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
94, 7, 83syl 18 . . . 4 (𝜑 → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
10 epweon 7133 . . . 4 E We On
11 wess 5237 . . . 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 2771 . . . 4 {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} = {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)}
1413f1owe 6748 . . 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 6344 . . . . . . . . 9 ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ∈ V
1716epelc 5165 . . . . . . . 8 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))
181, 2, 3dnnumch3lem 38142 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1918adantrr 696 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
201, 2, 3dnnumch3lem 38142 . . . . . . . . . 10 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2120adantrl 695 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2219, 21eleq12d 2844 . . . . . . . 8 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})))
2317, 22syl5rbb 273 . . . . . . 7 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ( (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)))
2423pm5.32da 568 . . . . . 6 (𝜑 → (((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})) ↔ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))))
2524opabbidv 4851 . . . . 5 (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))} = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))})
26 incom 3956 . . . . . 6 (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27 df-xp 5256 . . . . . . 7 (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)}
28 dnwech.h . . . . . . 7 𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}
2927, 28ineq12i 3963 . . . . . 6 ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})})
30 inopab 5390 . . . . . 6 ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
3126, 29, 303eqtri 2797 . . . . 5 (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
32 incom 3956 . . . . . 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 3961 . . . . . 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 5390 . . . . . 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 2797 . . . . 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 2830 . . . 4 (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37 weeq1 5238 . . . 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 5325 . . 3 (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40 weinxp 5325 . . 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 303 . 2 (𝜑 → (𝐻 We 𝐴 ↔ {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} We 𝐴))
4215, 41mpbird 247 1 (𝜑𝐻 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  cdif 3720  cin 3722  wss 3723  c0 4063  𝒫 cpw 4298  {csn 4317   cint 4612   class class class wbr 4787  {copab 4847  cmpt 4864   E cep 5162   We wwe 5208   × cxp 5248  ccnv 5249  ran crn 5251  cima 5253  Oncon0 5865  wf 6026  1-1wf1 6027  1-1-ontowf1o 6029  cfv 6030  recscrecs 7623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-wrecs 7562  df-recs 7624
This theorem is referenced by:  aomclem3  38152
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