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Theorem frpoinsg 32045
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 3731 . . . . . . . . 9 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑})
2 nfcv 2900 . . . . . . . . . . . 12 𝑦𝐴
32elrabsf 3613 . . . . . . . . . . 11 (𝑧 ∈ {𝑦𝐴𝜑} ↔ (𝑧𝐴[𝑧 / 𝑦]𝜑))
43simprbi 483 . . . . . . . . . 10 (𝑧 ∈ {𝑦𝐴𝜑} → [𝑧 / 𝑦]𝜑)
54ralimi 3088 . . . . . . . . 9 (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
61, 5sylbi 207 . . . . . . . 8 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
7 nfv 1990 . . . . . . . . . 10 𝑦((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴)
8 nfcv 2900 . . . . . . . . . . . 12 𝑦Pred(𝑅, 𝐴, 𝑤)
9 nfsbc1v 3594 . . . . . . . . . . . 12 𝑦[𝑧 / 𝑦]𝜑
108, 9nfral 3081 . . . . . . . . . . 11 𝑦𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
11 nfsbc1v 3594 . . . . . . . . . . 11 𝑦[𝑤 / 𝑦]𝜑
1210, 11nfim 1972 . . . . . . . . . 10 𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)
137, 12nfim 1972 . . . . . . . . 9 𝑦(((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
14 eleq1w 2820 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
1514anbi2d 742 . . . . . . . . . 10 (𝑦 = 𝑤 → (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) ↔ ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴)))
16 predeq3 5843 . . . . . . . . . . . 12 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1716raleqdv 3281 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18 sbceq1a 3585 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝜑[𝑤 / 𝑦]𝜑))
1917, 18imbi12d 333 . . . . . . . . . 10 (𝑦 = 𝑤 → ((∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑) ↔ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)))
2015, 19imbi12d 333 . . . . . . . . 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, 21chvar 2405 . . . . . . . 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 479 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → 𝑤𝐴)
2523, 24jctild 567 . . . . . 6 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → (𝑤𝐴[𝑤 / 𝑦]𝜑)))
262elrabsf 3613 . . . . . 6 (𝑤 ∈ {𝑦𝐴𝜑} ↔ (𝑤𝐴[𝑤 / 𝑦]𝜑))
2725, 26syl6ibr 242 . . . . 5 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑤𝐴) → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))
2827ralrimiva 3102 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))
29 ssrab2 3826 . . . 4 {𝑦𝐴𝜑} ⊆ 𝐴
3028, 29jctil 561 . . 3 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ({𝑦𝐴𝜑} ⊆ 𝐴 ∧ ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑})))
31 frpoind 32044 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ({𝑦𝐴𝜑} ⊆ 𝐴 ∧ ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))) → 𝐴 = {𝑦𝐴𝜑})
3230, 31mpdan 705 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐴 = {𝑦𝐴𝜑})
33 rabid2 3255 . 2 (𝐴 = {𝑦𝐴𝜑} ↔ ∀𝑦𝐴 𝜑)
3432, 33sylib 208 1 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  {crab 3052  [wsbc 3574  wss 3713   Po wpo 5183   Fr wfr 5220   Se wse 5221  Predcpred 5838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-po 5185  df-fr 5223  df-se 5224  df-xp 5270  df-cnv 5272  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839
This theorem is referenced by: (None)
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