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Theorem tz9.12 8513
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 8510 through tz9.12lem3 8512. (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 2609 . . . 4 (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
31, 2tz9.12lem2 8511 . . 3 suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) ∈ On
43onsuci 6907 . 2 suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) ∈ On
51, 2tz9.12lem3 8512 . 2 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴)))
6 fveq2 6088 . . . 4 (𝑦 = suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) → (𝑅1𝑦) = (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴)))
76eleq2d 2672 . . 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 3281 . 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 693 1 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2895  wrex 2896  {crab 2899  Vcvv 3172   cuni 4366   cint 4404  cmpt 4637  cima 5031  Oncon0 5626  suc csuc 5628  cfv 5790  𝑅1cr1 8485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-r1 8487
This theorem is referenced by:  tz9.13  8514  r1elss  8529
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