Theorem wfisg 5935

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. (Contributed by Scott Fenton, 11-Feb-2011.)

Hypothesis

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

Assertion

Ref	Expression
wfisg	⊢ ((𝑅 We 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 wfisg

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3891	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		dfss3 3794	. . . . . . . 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 3679	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜑))
5	4	simprbi 486	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → [𝑧 / 𝑦]𝜑)
6	5	ralimi 3147	. . . . . . . 8 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
7	2, 6	sylbi 208	. . . . . . 7 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8		nfv 2005	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
9		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑦Pred(𝑅, 𝐴, 𝑤)
10		nfsbc1v 3660	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
11	9, 10	nfral 3140	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12		nfsbc1v 3660	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
13	11, 12	nfim 1987	. . . . . . . . 9 ⊢ Ⅎ𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑)
14	8, 13	nfim 1987	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))
15		eleq1w 2875	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
16		predeq3 5904	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
17	16	raleqdv 3340	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18		sbceq1a 3651	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
19	17, 18	imbi12d 335	. . . . . . . . 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 335	. . . . . . . 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		wfisg.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))
22	14, 20, 21	chvar 2438	. . . . . . 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 547	. . . . 5 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑)))
25	3	elrabsf 3679	. . . . 5 ⊢ (𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑))
26	24, 25	syl6ibr 243	. . . 4 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}))
27	26	rgen 3117	. . 3 ⊢ ∀𝑤 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
28		wfi 5933	. . 3 ⊢ (((𝑅 We 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 688	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑})
30		rabid2 3314	. 2 ⊢ (𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝐴 𝜑)
31	29, 30	sylib 209	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1637   ∈ wcel 2157  ∀wral 3103  {crab 3107  [wsbc 3640   ⊆ wss 3776   Se wse 5275   We wwe 5276  Predcpred 5899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-cnv 5326  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900

This theorem is referenced by:  wfis  5936  wfis2fg  5937