Theorem findcard2 15745

Description: Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets.

Hypotheses

Ref	Expression
findcard2.1	|- (x = (/) -> (ph <-> ps))
findcard2.2	|- (x = y -> (ph <-> ch))
findcard2.3	|- (x = (y u. {z}) -> (ph <-> th))
findcard2.4	|- (x = A -> (ph <-> ta))
findcard2.5	|- ps
findcard2.6	|- (y e. Fin -> (ch -> th))

Assertion

Ref	Expression
findcard2	|- (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 findcard2

Step	Hyp	Ref	Expression
1		findcard2.4	. 2 |- (x = A -> (ph <-> ta))
2		isfi 5441	. . 3 |- (x e. Fin <-> E.w e. om x ~~ w)
3		breq2 3342	. . . . . . . 8 |- (w = (/) -> (x ~~ w <-> x ~~ (/)))
4	3	imbi1d 675	. . . . . . 7 |- (w = (/) -> ((x ~~ w -> ph) <-> (x ~~ (/) -> ph)))
5	4	albidv 1656	. . . . . 6 |- (w = (/) -> (A.x(x ~~ w -> ph) <-> A.x(x ~~ (/) -> ph)))
6		breq2 3342	. . . . . . . 8 |- (w = v -> (x ~~ w <-> x ~~ v))
7	6	imbi1d 675	. . . . . . 7 |- (w = v -> ((x ~~ w -> ph) <-> (x ~~ v -> ph)))
8	7	albidv 1656	. . . . . 6 |- (w = v -> (A.x(x ~~ w -> ph) <-> A.x(x ~~ v -> ph)))
9		breq2 3342	. . . . . . . 8 |- (w = suc v -> (x ~~ w <-> x ~~ suc v))
10	9	imbi1d 675	. . . . . . 7 |- (w = suc v -> ((x ~~ w -> ph) <-> (x ~~ suc v -> ph)))
11	10	albidv 1656	. . . . . 6 |- (w = suc v -> (A.x(x ~~ w -> ph) <-> A.x(x ~~ suc v -> ph)))
12		en0 5482	. . . . . . . 8 |- (x ~~ (/) <-> x = (/))
13		findcard2.5	. . . . . . . . 9 |- ps
14		findcard2.1	. . . . . . . . 9 |- (x = (/) -> (ph <-> ps))
15	13, 14	mpbiri 211	. . . . . . . 8 |- (x = (/) -> ph)
16	12, 15	sylbi 216	. . . . . . 7 |- (x ~~ (/) -> ph)
17	16	ax-gen 1305	. . . . . 6 |- A.x(x ~~ (/) -> ph)
18		nsuceq0 3749	. . . . . . . . . . . 12 |- suc v =/= (/)
19		breq1 3341	. . . . . . . . . . . . . . . 16 |- (w = (/) -> (w ~~ suc v <-> (/) ~~ suc v))
20	19	anbi2d 678	. . . . . . . . . . . . . . 15 |- (w = (/) -> ((v e. om /\ w ~~ suc v) <-> (v e. om /\ (/) ~~ suc v)))
21		nneneq 5606	. . . . . . . . . . . . . . . . . 18 |- (((/) e. om /\ suc v e. om) -> ((/) ~~ suc v <-> (/) = suc v))
22		peano1 3971	. . . . . . . . . . . . . . . . . 18 |- (/) e. om
23		peano2 3972	. . . . . . . . . . . . . . . . . 18 |- (v e. om -> suc v e. om)
24	21, 22, 23	sylancr 526	. . . . . . . . . . . . . . . . 17 |- (v e. om -> ((/) ~~ suc v <-> (/) = suc v))
25	24	biimpa 460	. . . . . . . . . . . . . . . 16 |- ((v e. om /\ (/) ~~ suc v) -> (/) = suc v)
26	25	eqcomd 1889	. . . . . . . . . . . . . . 15 |- ((v e. om /\ (/) ~~ suc v) -> suc v = (/))
27	20, 26	syl6bi 231	. . . . . . . . . . . . . 14 |- (w = (/) -> ((v e. om /\ w ~~ suc v) -> suc v = (/)))
28	27	com12 14	. . . . . . . . . . . . 13 |- ((v e. om /\ w ~~ suc v) -> (w = (/) -> suc v = (/)))
29	28	necon3d 2041	. . . . . . . . . . . 12 |- ((v e. om /\ w ~~ suc v) -> (suc v =/= (/) -> w =/= (/)))
30	18, 29	mpi 55	. . . . . . . . . . 11 |- ((v e. om /\ w ~~ suc v) -> w =/= (/))
31	30	ex 402	. . . . . . . . . 10 |- (v e. om -> (w ~~ suc v -> w =/= (/)))
32		visset 2295	. . . . . . . . . . . . . . . 16 |- w e. _V
33	32	dif1en 10172	. . . . . . . . . . . . . . 15 |- ((v e. om /\ w ~~ suc v /\ z e. w) -> (w \ {z}) ~~ v)
34	33	3expia 1069	. . . . . . . . . . . . . 14 |- ((v e. om /\ w ~~ suc v) -> (z e. w -> (w \ {z}) ~~ v))
35		snssi 3129	. . . . . . . . . . . . . . . . . 18 |- (z e. w -> {z} C_ w)
36		undif 2954	. . . . . . . . . . . . . . . . . . . 20 |- ({z} C_ w <-> ({z} u. (w \ {z})) = w)
37	36	biimpi 168	. . . . . . . . . . . . . . . . . . 19 |- ({z} C_ w -> ({z} u. (w \ {z})) = w)
38		uncom 2744	. . . . . . . . . . . . . . . . . . 19 |- ((w \ {z}) u. {z}) = ({z} u. (w \ {z}))
39	37, 38	syl5eq 1940	. . . . . . . . . . . . . . . . . 18 |- ({z} C_ w -> ((w \ {z}) u. {z}) = w)
40		dfsbcq 2455	. . . . . . . . . . . . . . . . . . . 20 |- (((w \ {z}) u. {z}) = w -> ([((w \ {z}) u. {z}) / x]ph <-> [w / x]ph))
41	40	imbi2d 674	. . . . . . . . . . . . . . . . . . 19 |- (((w \ {z}) u. {z}) = w -> ((A.x(x ~~ v -> ph) -> [((w \ {z}) u. {z}) / x]ph) <-> (A.x(x ~~ v -> ph) -> [w / x]ph)))
42		difexg 3458	. . . . . . . . . . . . . . . . . . . . 21 |- (w e. _V -> (w \ {z}) e. _V)
43	32, 42	ax-mp 7	. . . . . . . . . . . . . . . . . . . 20 |- (w \ {z}) e. _V
44		breq1 3341	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = (w \ {z}) -> (y ~~ v <-> (w \ {z}) ~~ v))
45	44	anbi2d 678	. . . . . . . . . . . . . . . . . . . . 21 |- (y = (w \ {z}) -> ((v e. om /\ y ~~ v) <-> (v e. om /\ (w \ {z}) ~~ v)))
46		uneq1 2748	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = (w \ {z}) -> (y u. {z}) = ((w \ {z}) u. {z}))
47		dfsbcq 2455	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y u. {z}) = ((w \ {z}) u. {z}) -> ([(y u. {z}) / x]ph <-> [((w \ {z}) u. {z}) / x]ph))
48	46, 47	syl 12	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = (w \ {z}) -> ([(y u. {z}) / x]ph <-> [((w \ {z}) u. {z}) / x]ph))
49	48	imbi2d 674	. . . . . . . . . . . . . . . . . . . . 21 |- (y = (w \ {z}) -> ((A.x(x ~~ v -> ph) -> [(y u. {z}) / x]ph) <-> (A.x(x ~~ v -> ph) -> [((w \ {z}) u. {z}) / x]ph)))
50	45, 49	imbi12d 688	. . . . . . . . . . . . . . . . . . . 20 |- (y = (w \ {z}) -> (((v e. om /\ y ~~ v) -> (A.x(x ~~ v -> ph) -> [(y u. {z}) / x]ph)) <-> ((v e. om /\ (w \ {z}) ~~ v) -> (A.x(x ~~ v -> ph) -> [((w \ {z}) u. {z}) / x]ph))))
51		ra4e 2156	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((v e. om /\ y ~~ v) -> E.v e. om y ~~ v)
52		isfi 5441	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. Fin <-> E.v e. om y ~~ v)
53	51, 52	sylibr 217	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((v e. om /\ y ~~ v) -> y e. Fin)
54		pm2.27 76	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y ~~ v -> ((y ~~ v -> ch) -> ch))
55	54	adantl 424	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((v e. om /\ y ~~ v) -> ((y ~~ v -> ch) -> ch))
56		findcard2.6	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. Fin -> (ch -> th))
57	53, 55, 56	sylsyld 32	. . . . . . . . . . . . . . . . . . . . . 22 |- ((v e. om /\ y ~~ v) -> ((y ~~ v -> ch) -> th))
58		visset 2295	. . . . . . . . . . . . . . . . . . . . . . 23 |- y e. _V
59		breq1 3341	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x = y -> (x ~~ v <-> y ~~ v))
60		findcard2.2	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x = y -> (ph <-> ch))
61	59, 60	imbi12d 688	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x = y -> ((x ~~ v -> ph) <-> (y ~~ v -> ch)))
62	58, 61	cla4v 2370	. . . . . . . . . . . . . . . . . . . . . 22 |- (A.x(x ~~ v -> ph) -> (y ~~ v -> ch))
63	57, 62	syl5 20	. . . . . . . . . . . . . . . . . . . . 21 |- ((v e. om /\ y ~~ v) -> (A.x(x ~~ v -> ph) -> th))
64		snex 3492	. . . . . . . . . . . . . . . . . . . . . . 23 |- {z} e. _V
65	58, 64	unex 3796	. . . . . . . . . . . . . . . . . . . . . 22 |- (y u. {z}) e. _V
66		findcard2.3	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = (y u. {z}) -> (ph <-> th))
67	65, 66	sbcie 2485	. . . . . . . . . . . . . . . . . . . . 21 |- ([(y u. {z}) / x]ph <-> th)
68	63, 67	syl6ibr 230	. . . . . . . . . . . . . . . . . . . 20 |- ((v e. om /\ y ~~ v) -> (A.x(x ~~ v -> ph) -> [(y u. {z}) / x]ph))
69	43, 50, 68	vtocl 2339	. . . . . . . . . . . . . . . . . . 19 |- ((v e. om /\ (w \ {z}) ~~ v) -> (A.x(x ~~ v -> ph) -> [((w \ {z}) u. {z}) / x]ph))
70	41, 69	syl5bi 225	. . . . . . . . . . . . . . . . . 18 |- (((w \ {z}) u. {z}) = w -> ((v e. om /\ (w \ {z}) ~~ v) -> (A.x(x ~~ v -> ph) -> [w / x]ph)))
71	35, 39, 70	3syl 24	. . . . . . . . . . . . . . . . 17 |- (z e. w -> ((v e. om /\ (w \ {z}) ~~ v) -> (A.x(x ~~ v -> ph) -> [w / x]ph)))
72	71	exp3a 405	. . . . . . . . . . . . . . . 16 |- (z e. w -> (v e. om -> ((w \ {z}) ~~ v -> (A.x(x ~~ v -> ph) -> [w / x]ph))))
73	72	com12 14	. . . . . . . . . . . . . . 15 |- (v e. om -> (z e. w -> ((w \ {z}) ~~ v -> (A.x(x ~~ v -> ph) -> [w / x]ph))))
74	73	adantr 425	. . . . . . . . . . . . . 14 |- ((v e. om /\ w ~~ suc v) -> (z e. w -> ((w \ {z}) ~~ v -> (A.x(x ~~ v -> ph) -> [w / x]ph))))
75	34, 74	mpdd 57	. . . . . . . . . . . . 13 |- ((v e. om /\ w ~~ suc v) -> (z e. w -> (A.x(x ~~ v -> ph) -> [w / x]ph)))
76	75	19.23adv 1584	. . . . . . . . . . . 12 |- ((v e. om /\ w ~~ suc v) -> (E.z z e. w -> (A.x(x ~~ v -> ph) -> [w / x]ph)))
77		n0 2884	. . . . . . . . . . . 12 |- (w =/= (/) <-> E.z z e. w)
78	76, 77	syl5ib 223	. . . . . . . . . . 11 |- ((v e. om /\ w ~~ suc v) -> (w =/= (/) -> (A.x(x ~~ v -> ph) -> [w / x]ph)))
79	78	ex 402	. . . . . . . . . 10 |- (v e. om -> (w ~~ suc v -> (w =/= (/) -> (A.x(x ~~ v -> ph) -> [w / x]ph))))
80	31, 79	mpdd 57	. . . . . . . . 9 |- (v e. om -> (w ~~ suc v -> (A.x(x ~~ v -> ph) -> [w / x]ph)))
81	80	com23 36	. . . . . . . 8 |- (v e. om -> (A.x(x ~~ v -> ph) -> (w ~~ suc v -> [w / x]ph)))
82	81	19.21adv 1666	. . . . . . 7 |- (v e. om -> (A.x(x ~~ v -> ph) -> A.w(w ~~ suc v -> [w / x]ph)))
83		ax-17 1317	. . . . . . . 8 |- ((x ~~ suc v -> ph) -> A.w(x ~~ suc v -> ph))
84		ax-17 1317	. . . . . . . . 9 |- (w ~~ suc v -> A.x w ~~ suc v)
85		ax-17 1317	. . . . . . . . . 10 |- (ph -> A.wph)
86	85	hbsb3 1575	. . . . . . . . 9 |- ([w / x]ph -> A.x[w / x]ph)
87	84, 86	hbim 1354	. . . . . . . 8 |- ((w ~~ suc v -> [w / x]ph) -> A.x(w ~~ suc v -> [w / x]ph))
88		breq1 3341	. . . . . . . . 9 |- (x = w -> (x ~~ suc v <-> w ~~ suc v))
89		sbequ12 1545	. . . . . . . . 9 |- (x = w -> (ph <-> [w / x]ph))
90	88, 89	imbi12d 688	. . . . . . . 8 |- (x = w -> ((x ~~ suc v -> ph) <-> (w ~~ suc v -> [w / x]ph)))
91	83, 87, 90	cbval 1527	. . . . . . 7 |- (A.x(x ~~ suc v -> ph) <-> A.w(w ~~ suc v -> [w / x]ph))
92	82, 91	syl6ibr 230	. . . . . 6 |- (v e. om -> (A.x(x ~~ v -> ph) -> A.x(x ~~ suc v -> ph)))
93	5, 8, 11, 17, 92	finds1 3982	. . . . 5 |- (w e. om -> A.x(x ~~ w -> ph))
94	93	19.21bi 1408	. . . 4 |- (w e. om -> (x ~~ w -> ph))
95	94	r19.23aiv 2211	. . 3 |- (E.w e. om x ~~ w -> ph)
96	2, 95	sylbi 216	. 2 |- (x e. Fin -> ph)
97	1, 96	vtoclga 2352	1 |- (A e. Fin -> ta)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534   =/= wne 2017  E.wrex 2106  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338  suc csuc 3659  omcom 3949   ~~ cen 5423  Fincfn 5426

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-sbc 2454  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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