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Theorem frpoinsg 33166
Description: Founded, Partial-Ordering Induction Schema. If a property passes from all elements less than 𝑦 of a founded, partially-ordered class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2022.)
Hypothesis
Ref Expression
frpoinsg.1 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
Assertion
Ref Expression
frpoinsg ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem frpoinsg
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dfss3 3941 . . . . . . . . 9 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑})
2 nfcv 2982 . . . . . . . . . . . 12 𝑦𝐴
32elrabsf 3802 . . . . . . . . . . 11 (𝑧 ∈ {𝑦𝐴𝜑} ↔ (𝑧𝐴[𝑧 / 𝑦]𝜑))
43simprbi 500 . . . . . . . . . 10 (𝑧 ∈ {𝑦𝐴𝜑} → [𝑧 / 𝑦]𝜑)
54ralimi 3155 . . . . . . . . 9 (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
61, 5sylbi 220 . . . . . . . 8 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
7 nfv 1916 . . . . . . . . . 10 𝑦((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴)
8 nfcv 2982 . . . . . . . . . . . 12 𝑦Pred(𝑅, 𝐴, 𝑤)
9 nfsbc1v 3778 . . . . . . . . . . . 12 𝑦[𝑧 / 𝑦]𝜑
108, 9nfralw 3219 . . . . . . . . . . 11 𝑦𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
11 nfsbc1v 3778 . . . . . . . . . . 11 𝑦[𝑤 / 𝑦]𝜑
1210, 11nfim 1898 . . . . . . . . . 10 𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)
137, 12nfim 1898 . . . . . . . . 9 𝑦(((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
14 eleq1w 2898 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
1514anbi2d 631 . . . . . . . . . 10 (𝑦 = 𝑤 → (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) ↔ ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴)))
16 predeq3 6141 . . . . . . . . . . . 12 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1716raleqdv 3402 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18 sbceq1a 3769 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝜑[𝑤 / 𝑦]𝜑))
1917, 18imbi12d 348 . . . . . . . . . 10 (𝑦 = 𝑤 → ((∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑) ↔ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)))
2015, 19imbi12d 348 . . . . . . . . 9 (𝑦 = 𝑤 → ((((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑)) ↔ (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))))
21 frpoinsg.1 . . . . . . . . 9 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
2213, 20, 21chvarfv 2244 . . . . . . . 8 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
236, 22syl5 34 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → [𝑤 / 𝑦]𝜑))
24 simpr 488 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → 𝑤𝐴)
2523, 24jctild 529 . . . . . 6 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → (𝑤𝐴[𝑤 / 𝑦]𝜑)))
262elrabsf 3802 . . . . . 6 (𝑤 ∈ {𝑦𝐴𝜑} ↔ (𝑤𝐴[𝑤 / 𝑦]𝜑))
2725, 26syl6ibr 255 . . . . 5 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))
2827ralrimiva 3177 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))
29 ssrab2 4042 . . . 4 {𝑦𝐴𝜑} ⊆ 𝐴
3028, 29jctil 523 . . 3 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ({𝑦𝐴𝜑} ⊆ 𝐴 ∧ ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑})))
31 frpoind 33165 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ({𝑦𝐴𝜑} ⊆ 𝐴 ∧ ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))) → 𝐴 = {𝑦𝐴𝜑})
3230, 31mpdan 686 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐴 = {𝑦𝐴𝜑})
33 rabid2 3372 . 2 (𝐴 = {𝑦𝐴𝜑} ↔ ∀𝑦𝐴 𝜑)
3432, 33sylib 221 1 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  {crab 3137  [wsbc 3758  wss 3919   Po wpo 5460   Fr wfr 5499   Se wse 5500  Predcpred 6136
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-po 5462  df-fr 5502  df-se 5503  df-xp 5549  df-cnv 5551  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137
This theorem is referenced by:  frpoins2fg  33167
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