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Theorem dnwech 41361
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 41360 . . . 4 (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
5 f1f1orn 6795 . . . 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 6738 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On)
8 frn 6675 . . . . 5 ((𝑥𝐴 (𝐹 “ {𝑥})):𝐴⟶On → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
94, 7, 83syl 18 . . . 4 (𝜑 → ran (𝑥𝐴 (𝐹 “ {𝑥})) ⊆ On)
10 epweon 7709 . . . 4 E We On
11 wess 5620 . . . 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 2736 . . . 4 {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} = {⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)}
1413f1owe 7298 . . 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 6855 . . . . . . . . 9 ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ∈ V
1716epeli 5539 . . . . . . . 8 (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))
181, 2, 3dnnumch3lem 41359 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
1918adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) = (𝐹 “ {𝑣}))
201, 2, 3dnnumch3lem 41359 . . . . . . . . . 10 ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2120adantrl 714 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
2219, 21eleq12d 2832 . . . . . . . 8 ((𝜑 ∧ (𝑣𝐴𝑤𝐴)) → (((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) ∈ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) ↔ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})))
2317, 22bitr2id 283 . . . . . . 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 5171 . . . . 5 (𝜑 → {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))} = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤))})
26 incom 4161 . . . . . 6 (𝐻 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝐻)
27 df-xp 5639 . . . . . . 7 (𝐴 × 𝐴) = {⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)}
28 dnwech.h . . . . . . 7 𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}
2927, 28ineq12i 4170 . . . . . 6 ((𝐴 × 𝐴) ∩ 𝐻) = ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})})
30 inopab 5785 . . . . . 6 ({⟨𝑣, 𝑤⟩ ∣ (𝑣𝐴𝑤𝐴)} ∩ {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
3126, 29, 303eqtri 2768 . . . . 5 (𝐻 ∩ (𝐴 × 𝐴)) = {⟨𝑣, 𝑤⟩ ∣ ((𝑣𝐴𝑤𝐴) ∧ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤}))}
32 incom 4161 . . . . . 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 4168 . . . . . 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 5785 . . . . . 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 2768 . . . . 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 2801 . . . 4 (𝜑 → (𝐻 ∩ (𝐴 × 𝐴)) = ({⟨𝑣, 𝑤⟩ ∣ ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑣) E ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤)} ∩ (𝐴 × 𝐴)))
37 weeq1 5621 . . . 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 5716 . . 3 (𝐻 We 𝐴 ↔ (𝐻 ∩ (𝐴 × 𝐴)) We 𝐴)
40 weinxp 5716 . . 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 313 . 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 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cint 4907   class class class wbr 5105  {copab 5167  cmpt 5188   E cep 5536   We wwe 5587   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636  Oncon0 6317  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  recscrecs 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317
This theorem is referenced by:  aomclem3  41369
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