Theorem frinsg 33200

Description: Founded Induction Schema. If a property passes from all elements less than 𝑦 of a founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypothesis

Ref	Expression
frinsg.1	⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))

Assertion

Ref	Expression
frinsg	⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		ssrab2 4007	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		dfss3 3903	. . . . . . . 8 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
3		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐴
4	3	elrabsf 3764	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜑))
5	4	simprbi 500	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → [𝑧 / 𝑦]𝜑)
6	5	ralimi 3128	. . . . . . . 8 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
7	2, 6	sylbi 220	. . . . . . 7 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
9		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑦Pred(𝑅, 𝐴, 𝑤)
10		nfsbc1v 3740	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
11	9, 10	nfralw 3189	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12		nfsbc1v 3740	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
13	11, 12	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑)
14	8, 13	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))
15		eleq1w 2872	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
16		predeq3 6120	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
17	16	raleqdv 3364	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18		sbceq1a 3731	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
19	17, 18	imbi12d 348	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑) ↔ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑)))
20	15, 19	imbi12d 348	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ↔ (𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))))
21		frinsg.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))
22	14, 20, 21	chvarfv 2240	. . . . . . 7 ⊢ (𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))
23	7, 22	syl5 34	. . . . . 6 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → [𝑤 / 𝑦]𝜑))
24	23	anc2li 559	. . . . 5 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑)))
25	3	elrabsf 3764	. . . . 5 ⊢ (𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑))
26	24, 25	syl6ibr 255	. . . 4 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}))
27	26	rgen 3116	. . 3 ⊢ ∀𝑤 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
28		frind 33198	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ({𝑦 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}))) → 𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑})
29	1, 27, 28	mpanr12 704	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → 𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑})
30		rabid2 3334	. 2 ⊢ (𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝐴 𝜑)
31	29, 30	sylib 221	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  [wsbc 3720   ⊆ wss 3881   Fr wfr 5475   Se wse 5476  Predcpred 6115

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-trpred 33170

This theorem is referenced by:  frins  33201  frins2fg  33202