Theorem findcard 10178

Description: Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
findcard.1	|- (x = (/) -> (ph <-> ps))
findcard.2	|- (x = (y \ {z}) -> (ph <-> ch))
findcard.3	|- (x = y -> (ph <-> th))
findcard.4	|- (x = A -> (ph <-> ta))
findcard.5	|- ps
findcard.6	|- (y e. Fin -> (A.z e. y ch -> th))

Assertion

Ref	Expression
findcard	|- (A e. Fin -> ta)

Distinct variable groups:   x,y,z,A   ps,x   ch,x   th,x   ta,x   ph,y,z

Proof of Theorem findcard

Step	Hyp	Ref	Expression
1		elex 2302	. 2 |- (A e. Fin -> E.x x = A)
2		isfi 5441	. . . . . 6 |- (x e. Fin <-> E.w e. om x ~~ w)
3		breq2 3342	. . . . . . . . . . 11 |- (w = (/) -> (x ~~ w <-> x ~~ (/)))
4	3	imbi1d 675	. . . . . . . . . 10 |- (w = (/) -> ((x ~~ w -> ph) <-> (x ~~ (/) -> ph)))
5	4	albidv 1656	. . . . . . . . 9 |- (w = (/) -> (A.x(x ~~ w -> ph) <-> A.x(x ~~ (/) -> ph)))
6		breq2 3342	. . . . . . . . . . 11 |- (w = v -> (x ~~ w <-> x ~~ v))
7	6	imbi1d 675	. . . . . . . . . 10 |- (w = v -> ((x ~~ w -> ph) <-> (x ~~ v -> ph)))
8	7	albidv 1656	. . . . . . . . 9 |- (w = v -> (A.x(x ~~ w -> ph) <-> A.x(x ~~ v -> ph)))
9		breq2 3342	. . . . . . . . . . 11 |- (w = suc v -> (x ~~ w <-> x ~~ suc v))
10	9	imbi1d 675	. . . . . . . . . 10 |- (w = suc v -> ((x ~~ w -> ph) <-> (x ~~ suc v -> ph)))
11	10	albidv 1656	. . . . . . . . 9 |- (w = suc v -> (A.x(x ~~ w -> ph) <-> A.x(x ~~ suc v -> ph)))
12		en0 5482	. . . . . . . . . . 11 |- (x ~~ (/) <-> x = (/))
13		findcard.5	. . . . . . . . . . . 12 |- ps
14		findcard.1	. . . . . . . . . . . 12 |- (x = (/) -> (ph <-> ps))
15	13, 14	mpbiri 211	. . . . . . . . . . 11 |- (x = (/) -> ph)
16	12, 15	sylbi 216	. . . . . . . . . 10 |- (x ~~ (/) -> ph)
17	16	ax-gen 1305	. . . . . . . . 9 |- A.x(x ~~ (/) -> ph)
18		breq2 3342	. . . . . . . . . . . . . . . . 17 |- (w = suc v -> (y ~~ w <-> y ~~ suc v))
19	18	rcla4ev 2381	. . . . . . . . . . . . . . . 16 |- ((suc v e. om /\ y ~~ suc v) -> E.w e. om y ~~ w)
20		peano2 3972	. . . . . . . . . . . . . . . 16 |- (v e. om -> suc v e. om)
21	19, 20	sylan 497	. . . . . . . . . . . . . . 15 |- ((v e. om /\ y ~~ suc v) -> E.w e. om y ~~ w)
22		isfi 5441	. . . . . . . . . . . . . . 15 |- (y e. Fin <-> E.w e. om y ~~ w)
23	21, 22	sylibr 217	. . . . . . . . . . . . . 14 |- ((v e. om /\ y ~~ suc v) -> y e. Fin)
24	23	3adant2 895	. . . . . . . . . . . . 13 |- ((v e. om /\ A.x(x ~~ v -> ph) /\ y ~~ suc v) -> y e. Fin)
25		visset 2295	. . . . . . . . . . . . . . . . . . . . 21 |- y e. _V
26	25	dif1en 10172	. . . . . . . . . . . . . . . . . . . 20 |- ((v e. om /\ y ~~ suc v /\ z e. y) -> (y \ {z}) ~~ v)
27	26	3expa 1067	. . . . . . . . . . . . . . . . . . 19 |- (((v e. om /\ y ~~ suc v) /\ z e. y) -> (y \ {z}) ~~ v)
28		pm2.27 76	. . . . . . . . . . . . . . . . . . 19 |- ((y \ {z}) ~~ v -> (((y \ {z}) ~~ v -> ch) -> ch))
29	27, 28	syl 12	. . . . . . . . . . . . . . . . . 18 |- (((v e. om /\ y ~~ suc v) /\ z e. y) -> (((y \ {z}) ~~ v -> ch) -> ch))
30		difexg 3458	. . . . . . . . . . . . . . . . . . . 20 |- (y e. _V -> (y \ {z}) e. _V)
31	25, 30	ax-mp 7	. . . . . . . . . . . . . . . . . . 19 |- (y \ {z}) e. _V
32		breq1 3341	. . . . . . . . . . . . . . . . . . . 20 |- (x = (y \ {z}) -> (x ~~ v <-> (y \ {z}) ~~ v))
33		findcard.2	. . . . . . . . . . . . . . . . . . . 20 |- (x = (y \ {z}) -> (ph <-> ch))
34	32, 33	imbi12d 688	. . . . . . . . . . . . . . . . . . 19 |- (x = (y \ {z}) -> ((x ~~ v -> ph) <-> ((y \ {z}) ~~ v -> ch)))
35	31, 34	cla4v 2370	. . . . . . . . . . . . . . . . . 18 |- (A.x(x ~~ v -> ph) -> ((y \ {z}) ~~ v -> ch))
36	29, 35	syl5 20	. . . . . . . . . . . . . . . . 17 |- (((v e. om /\ y ~~ suc v) /\ z e. y) -> (A.x(x ~~ v -> ph) -> ch))
37	36	r19.21adva 2182	. . . . . . . . . . . . . . . 16 |- ((v e. om /\ y ~~ suc v) -> (A.x(x ~~ v -> ph) -> A.z e. y ch))
38	37	imp 377	. . . . . . . . . . . . . . 15 |- (((v e. om /\ y ~~ suc v) /\ A.x(x ~~ v -> ph)) -> A.z e. y ch)
39	38	an1rs 547	. . . . . . . . . . . . . 14 |- (((v e. om /\ A.x(x ~~ v -> ph)) /\ y ~~ suc v) -> A.z e. y ch)
40	39	3impa 1062	. . . . . . . . . . . . 13 |- ((v e. om /\ A.x(x ~~ v -> ph) /\ y ~~ suc v) -> A.z e. y ch)
41		findcard.6	. . . . . . . . . . . . 13 |- (y e. Fin -> (A.z e. y ch -> th))
42	24, 40, 41	sylc 83	. . . . . . . . . . . 12 |- ((v e. om /\ A.x(x ~~ v -> ph) /\ y ~~ suc v) -> th)
43	42	3exp 1066	. . . . . . . . . . 11 |- (v e. om -> (A.x(x ~~ v -> ph) -> (y ~~ suc v -> th)))
44	43	19.21adv 1666	. . . . . . . . . 10 |- (v e. om -> (A.x(x ~~ v -> ph) -> A.y(y ~~ suc v -> th)))
45		breq1 3341	. . . . . . . . . . . 12 |- (x = y -> (x ~~ suc v <-> y ~~ suc v))
46		findcard.3	. . . . . . . . . . . 12 |- (x = y -> (ph <-> th))
47	45, 46	imbi12d 688	. . . . . . . . . . 11 |- (x = y -> ((x ~~ suc v -> ph) <-> (y ~~ suc v -> th)))
48	47	cbvalv 1696	. . . . . . . . . 10 |- (A.x(x ~~ suc v -> ph) <-> A.y(y ~~ suc v -> th))
49	44, 48	syl6ibr 230	. . . . . . . . 9 |- (v e. om -> (A.x(x ~~ v -> ph) -> A.x(x ~~ suc v -> ph)))
50	5, 8, 11, 17, 49	finds1 3982	. . . . . . . 8 |- (w e. om -> A.x(x ~~ w -> ph))
51	50	19.21bi 1408	. . . . . . 7 |- (w e. om -> (x ~~ w -> ph))
52	51	r19.23aiv 2211	. . . . . 6 |- (E.w e. om x ~~ w -> ph)
53	2, 52	sylbi 216	. . . . 5 |- (x e. Fin -> ph)
54		eleq1 1957	. . . . . 6 |- (x = A -> (x e. Fin <-> A e. Fin))
55		findcard.4	. . . . . 6 |- (x = A -> (ph <-> ta))
56	54, 55	imbi12d 688	. . . . 5 |- (x = A -> ((x e. Fin -> ph) <-> (A e. Fin -> ta)))
57	53, 56	mpbii 210	. . . 4 |- (x = A -> (A e. Fin -> ta))
58	57	com12 14	. . 3 |- (A e. Fin -> (x = A -> ta))
59	58	19.23adv 1584	. 2 |- (A e. Fin -> (E.x x = A -> ta))
60	1, 59	mpd 29	1 |- (A e. Fin -> ta)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106  _Vcvv 2292   \ cdif 2590  (/)c0 2875  {csn 3044   class class class wbr 3338  suc csuc 3659  omcom 3949   ~~ cen 5423  Fincfn 5426

This theorem is referenced by:  fixp 10180  fimax 15746  inficl 15757  frfi 15771

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-er 5318  df-en 5427  df-fin 5430

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