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Theorem tz9.12 9245
 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 9242 through tz9.12lem3 9244. (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 2759 . . . 4 (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})
31, 2tz9.12lem2 9243 . . 3 suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) ∈ On
43onsuci 7553 . 2 suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) ∈ On
51, 2tz9.12lem3 9244 . 2 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴)))
6 fveq2 6659 . . . 4 (𝑦 = suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴) → (𝑅1𝑦) = (𝑅1‘suc suc ((𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)}) “ 𝐴)))
76eleq2d 2838 . . 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 3542 . 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 591 1 (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  {crab 3075  Vcvv 3410  ∪ cuni 4799  ∩ cint 4839   ↦ cmpt 5113   “ cima 5528  Oncon0 6170  suc csuc 6172  ‘cfv 6336  𝑅1cr1 9217 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-r1 9219 This theorem is referenced by:  tz9.13  9246  r1elss  9261
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