Theorem zfz1iso 11063

Description: A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)

Assertion

Ref	Expression
zfz1iso	|- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Distinct variable group:   A, f

Proof of Theorem zfz1iso

Dummy variables n x a k m w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 6912	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 120	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	adantl 277	. 2 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. n e. om A ~~ n )
4		simprlr 538	. . . 4 |- ( ( n e. om /\ ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) -> A e. Fin )
5		breq2 4087	. . . . . . . . 9 |- ( w = (/) -> ( x ~~ w <-> x ~~ (/) ) )
6	5	anbi2d 464	. . . . . . . 8 |- ( w = (/) -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) ) )
7	6	imbi1d 231	. . . . . . 7 |- ( w = (/) -> ( ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
8	7	albidv 1870	. . . . . 6 |- ( w = (/) -> ( A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
9		breq2 4087	. . . . . . . . 9 |- ( w = k -> ( x ~~ w <-> x ~~ k ) )
10	9	anbi2d 464	. . . . . . . 8 |- ( w = k -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) ) )
11	10	imbi1d 231	. . . . . . 7 |- ( w = k -> ( ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
12	11	albidv 1870	. . . . . 6 |- ( w = k -> ( A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
13		breq2 4087	. . . . . . . . 9 |- ( w = suc k -> ( x ~~ w <-> x ~~ suc k ) )
14	13	anbi2d 464	. . . . . . . 8 |- ( w = suc k -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) )
15	14	imbi1d 231	. . . . . . 7 |- ( w = suc k -> ( ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
16	15	albidv 1870	. . . . . 6 |- ( w = suc k -> ( A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
17		breq2 4087	. . . . . . . . 9 |- ( w = n -> ( x ~~ w <-> x ~~ n ) )
18	17	anbi2d 464	. . . . . . . 8 |- ( w = n -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) ) )
19	18	imbi1d 231	. . . . . . 7 |- ( w = n -> ( ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
20	19	albidv 1870	. . . . . 6 |- ( w = n -> ( A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
21		iso0 5941	. . . . . . . . . 10 |- (/) Isom < , < ( (/) , (/) )
22		en0 6947	. . . . . . . . . . . . . . . . 17 |- ( x ~~ (/) <-> x = (/) )
23	22	biimpi 120	. . . . . . . . . . . . . . . 16 |- ( x ~~ (/) -> x = (/) )
24	23	fveq2d 5631	. . . . . . . . . . . . . . 15 |- ( x ~~ (/) -> (♯ ` x ) = (♯ ` (/) ) )
25		hash0 11018	. . . . . . . . . . . . . . 15 |- (♯ ` (/) ) = 0
26	24, 25	eqtrdi 2278	. . . . . . . . . . . . . 14 |- ( x ~~ (/) -> (♯ ` x ) = 0 )
27	26	oveq2d 6017	. . . . . . . . . . . . 13 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = ( 1 ... 0 ) )
28		fz10 10242	. . . . . . . . . . . . 13 |- ( 1 ... 0 ) = (/)
29	27, 28	eqtrdi 2278	. . . . . . . . . . . 12 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = (/) )
30		isoeq4 5928	. . . . . . . . . . . 12 |- ( ( 1 ... (♯ ` x ) ) = (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
31	29, 30	syl 14	. . . . . . . . . . 11 |- ( x ~~ (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
32		isoeq5 5929	. . . . . . . . . . . 12 |- ( x = (/) -> ( (/) Isom < , < ( (/) , x ) <-> (/) Isom < , < ( (/) , (/) ) ) )
33	23, 32	syl 14	. . . . . . . . . . 11 |- ( x ~~ (/) -> ( (/) Isom < , < ( (/) , x ) <-> (/) Isom < , < ( (/) , (/) ) ) )
34	31, 33	bitrd 188	. . . . . . . . . 10 |- ( x ~~ (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , (/) ) ) )
35	21, 34	mpbiri 168	. . . . . . . . 9 |- ( x ~~ (/) -> (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
36		0ex 4211	. . . . . . . . . 10 |- (/) e. _V
37		isoeq1 5925	. . . . . . . . . 10 |- ( f = (/) -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
38	36, 37	spcev 2898	. . . . . . . . 9 |- ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
39	35, 38	syl 14	. . . . . . . 8 |- ( x ~~ (/) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
40	39	adantl 277	. . . . . . 7 |- ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
41	40	ax-gen 1495	. . . . . 6 |- A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
42		sseq1 3247	. . . . . . . . . . 11 |- ( a = x -> ( a C_ ZZ <-> x C_ ZZ ) )
43		eleq1 2292	. . . . . . . . . . 11 |- ( a = x -> ( a e. Fin <-> x e. Fin ) )
44	42, 43	anbi12d 473	. . . . . . . . . 10 |- ( a = x -> ( ( a C_ ZZ /\ a e. Fin ) <-> ( x C_ ZZ /\ x e. Fin ) ) )
45		breq1 4086	. . . . . . . . . 10 |- ( a = x -> ( a ~~ k <-> x ~~ k ) )
46	44, 45	anbi12d 473	. . . . . . . . 9 |- ( a = x -> ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) ) )
47		fveq2 5627	. . . . . . . . . . . . 13 |- ( a = x -> (♯ ` a ) = (♯ ` x ) )
48	47	oveq2d 6017	. . . . . . . . . . . 12 |- ( a = x -> ( 1 ... (♯ ` a ) ) = ( 1 ... (♯ ` x ) ) )
49		isoeq4 5928	. . . . . . . . . . . 12 |- ( ( 1 ... (♯ ` a ) ) = ( 1 ... (♯ ` x ) ) -> ( f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) <-> f Isom < , < ( ( 1 ... (♯ ` x ) ) , a ) ) )
50	48, 49	syl 14	. . . . . . . . . . 11 |- ( a = x -> ( f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) <-> f Isom < , < ( ( 1 ... (♯ ` x ) ) , a ) ) )
51		isoeq5 5929	. . . . . . . . . . 11 |- ( a = x -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , a ) <-> f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
52	50, 51	bitrd 188	. . . . . . . . . 10 |- ( a = x -> ( f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) <-> f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
53	52	exbidv 1871	. . . . . . . . 9 |- ( a = x -> ( E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) <-> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
54	46, 53	imbi12d 234	. . . . . . . 8 |- ( a = x -> ( ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) <-> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
55	54	cbvalv 1964	. . . . . . 7 |- ( A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) <-> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
56		simprll 537	. . . . . . . . . . . . 13 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> x C_ ZZ )
57		zssq 9822	. . . . . . . . . . . . 13 |- ZZ C_ QQ
58	56, 57	sstrdi 3236	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> x C_ QQ )
59		simprlr 538	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> x e. Fin )
60		vex 2802	. . . . . . . . . . . . . . . 16 |- k e. _V
61		nsuceq0g 4509	. . . . . . . . . . . . . . . 16 |- ( k e. _V -> suc k =/= (/) )
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 |- suc k =/= (/)
63	62	neii 2402	. . . . . . . . . . . . . 14 |- -. suc k = (/)
64		simplrr 536	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ x = (/) ) -> x ~~ suc k )
65	64	ensymd 6935	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ x = (/) ) -> suc k ~~ x )
66		simpr 110	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ x = (/) ) -> x = (/) )
67	65, 66	breqtrd 4109	. . . . . . . . . . . . . . . 16 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ x = (/) ) -> suc k ~~ (/) )
68		en0 6947	. . . . . . . . . . . . . . . 16 |- ( suc k ~~ (/) <-> suc k = (/) )
69	67, 68	sylib 122	. . . . . . . . . . . . . . 15 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ x = (/) ) -> suc k = (/) )
70	69	ex 115	. . . . . . . . . . . . . 14 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> ( x = (/) -> suc k = (/) ) )
71	63, 70	mtoi 668	. . . . . . . . . . . . 13 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> -. x = (/) )
72	71	neqned 2407	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> x =/= (/) )
73		fimaxq 11049	. . . . . . . . . . . 12 |- ( ( x C_ QQ /\ x e. Fin /\ x =/= (/) ) -> E. m e. x A. z e. x z <_ m )
74	58, 59, 72, 73	syl3anc 1271	. . . . . . . . . . 11 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> E. m e. x A. z e. x z <_ m )
75		simplll 533	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> k e. om )
76		simpllr 534	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) )
77	56	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> x C_ ZZ )
78	59	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> x e. Fin )
79		simplrr 536	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> x ~~ suc k )
80		simprl 529	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> m e. x )
81		simprr 531	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> A. z e. x z <_ m )
82	75, 76, 77, 78, 79, 80, 81	zfz1isolem1 11062	. . . . . . . . . . 11 |- ( ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) /\ ( m e. x /\ A. z e. x z <_ m ) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
83	74, 82	rexlimddv 2653	. . . . . . . . . 10 |- ( ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) /\ ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
84	83	ex 115	. . . . . . . . 9 |- ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
85	84	alrimiv 1920	. . . . . . . 8 |- ( ( k e. om /\ A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) ) -> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
86	85	ex 115	. . . . . . 7 |- ( k e. om -> ( A. a ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) ) -> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
87	55, 86	biimtrrid 153	. . . . . 6 |- ( k e. om -> ( A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) -> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) ) )
88	8, 12, 16, 20, 41, 87	finds 4692	. . . . 5 |- ( n e. om -> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
89	88	adantr 276	. . . 4 |- ( ( n e. om /\ ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) -> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
90		simpr 110	. . . 4 |- ( ( n e. om /\ ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) -> ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) )
91		sseq1 3247	. . . . . . . 8 |- ( x = A -> ( x C_ ZZ <-> A C_ ZZ ) )
92		eleq1 2292	. . . . . . . 8 |- ( x = A -> ( x e. Fin <-> A e. Fin ) )
93	91, 92	anbi12d 473	. . . . . . 7 |- ( x = A -> ( ( x C_ ZZ /\ x e. Fin ) <-> ( A C_ ZZ /\ A e. Fin ) ) )
94		breq1 4086	. . . . . . 7 |- ( x = A -> ( x ~~ n <-> A ~~ n ) )
95	93, 94	anbi12d 473	. . . . . 6 |- ( x = A -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) <-> ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) )
96		fveq2 5627	. . . . . . . . . 10 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
97	96	oveq2d 6017	. . . . . . . . 9 |- ( x = A -> ( 1 ... (♯ ` x ) ) = ( 1 ... (♯ ` A ) ) )
98		isoeq4 5928	. . . . . . . . 9 |- ( ( 1 ... (♯ ` x ) ) = ( 1 ... (♯ ` A ) ) -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> f Isom < , < ( ( 1 ... (♯ ` A ) ) , x ) ) )
99	97, 98	syl 14	. . . . . . . 8 |- ( x = A -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> f Isom < , < ( ( 1 ... (♯ ` A ) ) , x ) ) )
100		isoeq5 5929	. . . . . . . 8 |- ( x = A -> ( f Isom < , < ( ( 1 ... (♯ ` A ) ) , x ) <-> f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) )
101	99, 100	bitrd 188	. . . . . . 7 |- ( x = A -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) )
102	101	exbidv 1871	. . . . . 6 |- ( x = A -> ( E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) )
103	95, 102	imbi12d 234	. . . . 5 |- ( x = A -> ( ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) <-> ( ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) )
104	103	spcgv 2890	. . . 4 |- ( A e. Fin -> ( A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) -> ( ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) )
105	4, 89, 90, 104	syl3c 63	. . 3 |- ( ( n e. om /\ ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
106	105	an12s 565	. 2 |- ( ( ( A C_ ZZ /\ A e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
107	3, 106	rexlimddv 2653	1 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   =/= wne 2400   A.wral 2508   E.wrex 2509   _Vcvv 2799   C_ wss 3197   (/)c0 3491   class class class wbr 4083   suc csuc 4456   omcom 4682   ` cfv 5318   Isom wiso 5319  (class class class)co 6001   ~~ cen 6885   Fincfn 6887   0cc0 7999   1c1 8000   < clt 8181   <_ cle 8182   ZZcz 9446   QQcq 9814   ...cfz 10204  ♯chash 10997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-ihash 10998

This theorem is referenced by:  summodclem2  11893  zsumdc  11895  prodmodclem2  12088  zproddc  12090