MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frinsg Structured version   Visualization version   GIF version

Theorem frinsg 9645
Description: Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. Theorem 5.6(ii) of [Levy] p. 64. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
frinsg.1 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
Assertion
Ref Expression
frinsg ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem frinsg
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 4035 . . 3 {𝑦𝐴𝜑} ⊆ 𝐴
2 dfss3 3930 . . . . . . . 8 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑})
3 nfcv 2905 . . . . . . . . . . 11 𝑦𝐴
43elrabsf 3785 . . . . . . . . . 10 (𝑧 ∈ {𝑦𝐴𝜑} ↔ (𝑧𝐴[𝑧 / 𝑦]𝜑))
54simprbi 497 . . . . . . . . 9 (𝑧 ∈ {𝑦𝐴𝜑} → [𝑧 / 𝑦]𝜑)
65ralimi 3084 . . . . . . . 8 (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
72, 6sylbi 216 . . . . . . 7 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8 nfv 1917 . . . . . . . . 9 𝑦 𝑤𝐴
9 nfcv 2905 . . . . . . . . . . 11 𝑦Pred(𝑅, 𝐴, 𝑤)
10 nfsbc1v 3757 . . . . . . . . . . 11 𝑦[𝑧 / 𝑦]𝜑
119, 10nfralw 3292 . . . . . . . . . 10 𝑦𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12 nfsbc1v 3757 . . . . . . . . . 10 𝑦[𝑤 / 𝑦]𝜑
1311, 12nfim 1899 . . . . . . . . 9 𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)
148, 13nfim 1899 . . . . . . . 8 𝑦(𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
15 eleq1w 2820 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
16 predeq3 6255 . . . . . . . . . . 11 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1716raleqdv 3311 . . . . . . . . . 10 (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18 sbceq1a 3748 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝜑[𝑤 / 𝑦]𝜑))
1917, 18imbi12d 344 . . . . . . . . 9 (𝑦 = 𝑤 → ((∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑) ↔ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)))
2015, 19imbi12d 344 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑)) ↔ (𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))))
21 frinsg.1 . . . . . . . 8 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
2214, 20, 21chvarfv 2233 . . . . . . 7 (𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
237, 22syl5 34 . . . . . 6 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → [𝑤 / 𝑦]𝜑))
2423anc2li 556 . . . . 5 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → (𝑤𝐴[𝑤 / 𝑦]𝜑)))
253elrabsf 3785 . . . . 5 (𝑤 ∈ {𝑦𝐴𝜑} ↔ (𝑤𝐴[𝑤 / 𝑦]𝜑))
2624, 25syl6ibr 251 . . . 4 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))
2726rgen 3064 . . 3 𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑})
28 frind 9644 . . 3 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ ({𝑦𝐴𝜑} ⊆ 𝐴 ∧ ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))) → 𝐴 = {𝑦𝐴𝜑})
291, 27, 28mpanr12 703 . 2 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → 𝐴 = {𝑦𝐴𝜑})
30 rabid2 3434 . 2 (𝐴 = {𝑦𝐴𝜑} ↔ ∀𝑦𝐴 𝜑)
3129, 30sylib 217 1 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  {crab 3405  [wsbc 3737  wss 3908   Fr wfr 5583   Se wse 5584  Predcpred 6250
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 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664  ax-inf2 9535
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 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-oadd 8408  df-ttrcl 9602
This theorem is referenced by:  frins  9646  frins2f  9647
  Copyright terms: Public domain W3C validator