Theorem wfisg 13917

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.

Hypothesis

Ref	Expression
wfisg.1	|- (y e. A -> (A.z e. Pred (R, A, y)[z / y]ph -> ph))

Assertion

Ref	Expression
wfisg	|- ((R We A /\ A.x e. A Pred(R, A, x) e. _V) -> A.y e. A ph)

Distinct variable groups:   x,A,y   z,A   ph,x   ph,z   x,R,y   z,R,y

Proof of Theorem wfisg

Step	Hyp	Ref	Expression
1		ssrab2 2692	. . 3 |- {y e. A | ph} C_ A
2		ax-17 1317	. . . . . . . . 9 |- (w e. A -> A.y w e. A)
3		ax-17 1317	. . . . . . . . . . 11 |- (z e. Pred(R, A, w) -> A.y z e. Pred(R, A, w))
4		hbs1 1722	. . . . . . . . . . 11 |- ([z / y]ph -> A.y[z / y]ph)
5	3, 4	hbral 2146	. . . . . . . . . 10 |- (A.z e. Pred (R, A, w)[z / y]ph -> A.yA.z e. Pred (R, A, w)[z / y]ph)
6		hbs1 1722	. . . . . . . . . 10 |- ([w / y]ph -> A.y[w / y]ph)
7	5, 6	hbim 1354	. . . . . . . . 9 |- ((A.z e. Pred (R, A, w)[z / y]ph -> [w / y]ph) -> A.y(A.z e. Pred (R, A, w)[z / y]ph -> [w / y]ph))
8	2, 7	hbim 1354	. . . . . . . 8 |- ((w e. A -> (A.z e. Pred (R, A, w)[z / y]ph -> [w / y]ph)) -> A.y(w e. A -> (A.z e. Pred (R, A, w)[z / y]ph -> [w / y]ph)))
9		eleq1 1957	. . . . . . . . 9 |- (y = w -> (y e. A <-> w e. A))
10		predeq3 13883	. . . . . . . . . . 11 |- (y = w -> Pred(R, A, y) = Pred(R, A, w))
11	10	raleqdv 2269	. . . . . . . . . 10 |- (y = w -> (A.z e. Pred (R, A, y)[z / y]ph <-> A.z e. Pred (R, A, w)[z / y]ph))
12		sbequ12 1545	. . . . . . . . . 10 |- (y = w -> (ph <-> [w / y]ph))
13	11, 12	imbi12d 688	. . . . . . . . 9 |- (y = w -> ((A.z e. Pred (R, A, y)[z / y]ph -> ph) <-> (A.z e. Pred (R, A, w)[z / y]ph -> [w / y]ph)))
14	9, 13	imbi12d 688	. . . . . . . 8 |- (y = w -> ((y e. A -> (A.z e. Pred (R, A, y)[z / y]ph -> ph)) <-> (w e. A -> (A.z e. Pred (R, A, w)[z / y]ph -> [w / y]ph))))
15		wfisg.1	. . . . . . . 8 |- (y e. A -> (A.z e. Pred (R, A, y)[z / y]ph -> ph))
16	8, 14, 15	chvar 1530	. . . . . . 7 |- (w e. A -> (A.z e. Pred (R, A, w)[z / y]ph -> [w / y]ph))
17		dfss3 2611	. . . . . . . 8 |- (Pred(R, A, w) C_ {y e. A | ph} <-> A.z e. Pred (R, A, w)z e. {y e. A | ph})
18	2	elrabsf 2486	. . . . . . . . . 10 |- (z e. {y e. A | ph} <-> (z e. A /\ [z / y]ph))
19	18	simprbi 353	. . . . . . . . 9 |- (z e. {y e. A | ph} -> [z / y]ph)
20	19	ralimi 2168	. . . . . . . 8 |- (A.z e. Pred (R, A, w)z e. {y e. A | ph} -> A.z e. Pred (R, A, w)[z / y]ph)
21	17, 20	sylbi 216	. . . . . . 7 |- (Pred(R, A, w) C_ {y e. A | ph} -> A.z e. Pred (R, A, w)[z / y]ph)
22	16, 21	syl5 20	. . . . . 6 |- (w e. A -> (Pred(R, A, w) C_ {y e. A | ph} -> [w / y]ph))
23	22	anc2li 326	. . . . 5 |- (w e. A -> (Pred(R, A, w) C_ {y e. A | ph} -> (w e. A /\ [w / y]ph)))
24	2	elrabsf 2486	. . . . 5 |- (w e. {y e. A | ph} <-> (w e. A /\ [w / y]ph))
25	23, 24	syl6ibr 230	. . . 4 |- (w e. A -> (Pred(R, A, w) C_ {y e. A | ph} -> w e. {y e. A | ph}))
26	25	rgen 2159	. . 3 |- A.w e. A (Pred(R, A, w) C_ {y e. A | ph} -> w e. {y e. A | ph})
27		wfi 13915	. . 3 |- (((R We A /\ A.x e. A Pred(R, A, x) e. _V) /\ ({y e. A | ph} C_ A /\ A.w e. A (Pred(R, A, w) C_ {y e. A | ph} -> w e. {y e. A | ph}))) -> A = {y e. A | ph})
28	1, 26, 27	mpanr12 778	. 2 |- ((R We A /\ A.x e. A Pred(R, A, x) e. _V) -> A = {y e. A | ph})
29		rabid2 2254	. 2 |- (A = {y e. A | ph} <-> A.y e. A ph)
30	28, 29	sylib 215	1 |- ((R We A /\ A.x e. A Pred(R, A, x) e. _V) -> A.y e. A ph)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  {crab 2108  _Vcvv 2292   C_ wss 2593   We wwe 3624  Predcpred 13879

This theorem is referenced by:  wfis 13918  wfis2fg 13919

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-pred 13880

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