Theorem frinsg 32605

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 3947	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		dfss3 3848	. . . . . . . 8 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
3		nfcv 2933	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐴
4	3	elrabsf 3721	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜑))
5	4	simprbi 489	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → [𝑧 / 𝑦]𝜑)
6	5	ralimi 3111	. . . . . . . 8 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
7	2, 6	sylbi 209	. . . . . . 7 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8		nfv 1873	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
9		nfcv 2933	. . . . . . . . . . 11 ⊢ Ⅎ𝑦Pred(𝑅, 𝐴, 𝑤)
10		nfsbc1v 3702	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
11	9, 10	nfral 3175	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12		nfsbc1v 3702	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
13	11, 12	nfim 1859	. . . . . . . . 9 ⊢ Ⅎ𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑)
14	8, 13	nfim 1859	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))
15		eleq1w 2849	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
16		predeq3 5990	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
17	16	raleqdv 3356	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18		sbceq1a 3693	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
19	17, 18	imbi12d 337	. . . . . . . . 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 337	. . . . . . . 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	chvar 2326	. . . . . . 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 548	. . . . 5 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑)))
25	3	elrabsf 3721	. . . . 5 ⊢ (𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑))
26	24, 25	syl6ibr 244	. . . 4 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}))
27	26	rgen 3099	. . 3 ⊢ ∀𝑤 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
28		frind 32603	. . 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 692	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → 𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑})
30		rabid2 3321	. 2 ⊢ (𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝐴 𝜑)
31	29, 30	sylib 210	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∀wral 3089  {crab 3093  [wsbc 3682   ⊆ wss 3830   Fr wfr 5363   Se wse 5364  Predcpred 5985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-trpred 32575

This theorem is referenced by:  frins  32606  frins2fg  32607