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Theorem tz9.12 8691
Description: A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 8688 through tz9.12lem3 8690. (Contributed by NM, 22-Sep-2003.)
Hypothesis
Ref Expression
tz9.12.1 𝐴 ∈ V
Assertion
Ref Expression
tz9.12 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem tz9.12
Dummy variables 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tz9.12.1 . . . 4 𝐴 ∈ V
2 eqid 2651 . . . 4 (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
31, 2tz9.12lem2 8689 . . 3 suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) ∈ On
43onsuci 7080 . 2 suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) ∈ On
51, 2tz9.12lem3 8690 . 2 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴)))
6 fveq2 6229 . . . 4 (𝑦 = suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) → (𝑅1𝑦) = (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴)))
76eleq2d 2716 . . 3 (𝑦 = suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) → (𝐴 ∈ (𝑅1𝑦) ↔ 𝐴 ∈ (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴))))
87rspcev 3340 . 2 ((suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) ∈ On ∧ 𝐴 ∈ (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴))) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))
94, 5, 8sylancr 696 1 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231   cuni 4468   cint 4507  cmpt 4762  cima 5146  Oncon0 5761  suc csuc 5763  cfv 5926  𝑅1cr1 8663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665
This theorem is referenced by:  tz9.13  8692  r1elss  8707
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