Theorem zfz1iso 10950

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 6829	. . . 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 4038	. . . . . . . . 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 1838	. . . . . 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 4038	. . . . . . . . 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 1838	. . . . . 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 4038	. . . . . . . . 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 1838	. . . . . 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 4038	. . . . . . . . 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 1838	. . . . . 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 5867	. . . . . . . . . 10 |- (/) Isom < , < ( (/) , (/) )
22		en0 6863	. . . . . . . . . . . . . . . . 17 |- ( x ~~ (/) <-> x = (/) )
23	22	biimpi 120	. . . . . . . . . . . . . . . 16 |- ( x ~~ (/) -> x = (/) )
24	23	fveq2d 5565	. . . . . . . . . . . . . . 15 |- ( x ~~ (/) -> (♯ ` x ) = (♯ ` (/) ) )
25		hash0 10905	. . . . . . . . . . . . . . 15 |- (♯ ` (/) ) = 0
26	24, 25	eqtrdi 2245	. . . . . . . . . . . . . 14 |- ( x ~~ (/) -> (♯ ` x ) = 0 )
27	26	oveq2d 5941	. . . . . . . . . . . . 13 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = ( 1 ... 0 ) )
28		fz10 10138	. . . . . . . . . . . . 13 |- ( 1 ... 0 ) = (/)
29	27, 28	eqtrdi 2245	. . . . . . . . . . . 12 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = (/) )
30		isoeq4 5854	. . . . . . . . . . . 12 |- ( ( 1 ... (♯ ` x ) ) = (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
31	29, 30	syl 14	. . . . . . . . . . 11 |- ( x ~~ (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
32		isoeq5 5855	. . . . . . . . . . . 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 4161	. . . . . . . . . 10 |- (/) e. _V
37		isoeq1 5851	. . . . . . . . . 10 |- ( f = (/) -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
38	36, 37	spcev 2859	. . . . . . . . 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 1463	. . . . . 6 |- A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
42		sseq1 3207	. . . . . . . . . . 11 |- ( a = x -> ( a C_ ZZ <-> x C_ ZZ ) )
43		eleq1 2259	. . . . . . . . . . 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 4037	. . . . . . . . . 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 5561	. . . . . . . . . . . . 13 |- ( a = x -> (♯ ` a ) = (♯ ` x ) )
48	47	oveq2d 5941	. . . . . . . . . . . 12 |- ( a = x -> ( 1 ... (♯ ` a ) ) = ( 1 ... (♯ ` x ) ) )
49		isoeq4 5854	. . . . . . . . . . . 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 5855	. . . . . . . . . . 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 1839	. . . . . . . . 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 1932	. . . . . . 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 9718	. . . . . . . . . . . . 13 |- ZZ C_ QQ
58	56, 57	sstrdi 3196	. . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 16 |- k e. _V
61		nsuceq0g 4454	. . . . . . . . . . . . . . . 16 |- ( k e. _V -> suc k =/= (/) )
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 |- suc k =/= (/)
63	62	neii 2369	. . . . . . . . . . . . . 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 6851	. . . . . . . . . . . . . . . . 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 4060	. . . . . . . . . . . . . . . 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 6863	. . . . . . . . . . . . . . . 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 665	. . . . . . . . . . . . 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 2374	. . . . . . . . . . . 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 10936	. . . . . . . . . . . 12 |- ( ( x C_ QQ /\ x e. Fin /\ x =/= (/) ) -> E. m e. x A. z e. x z <_ m )
74	58, 59, 72, 73	syl3anc 1249	. . . . . . . . . . 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 10949	. . . . . . . . . . 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 2619	. . . . . . . . . 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 1888	. . . . . . . 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 4637	. . . . 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 3207	. . . . . . . 8 |- ( x = A -> ( x C_ ZZ <-> A C_ ZZ ) )
92		eleq1 2259	. . . . . . . 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 4037	. . . . . . 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 5561	. . . . . . . . . 10 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
97	96	oveq2d 5941	. . . . . . . . 9 |- ( x = A -> ( 1 ... (♯ ` x ) ) = ( 1 ... (♯ ` A ) ) )
98		isoeq4 5854	. . . . . . . . 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 5855	. . . . . . . 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 1839	. . . . . 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 2851	. . . 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 2619	1 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1506   e. wcel 2167   =/= wne 2367   A.wral 2475   E.wrex 2476   _Vcvv 2763   C_ wss 3157   (/)c0 3451   class class class wbr 4034   suc csuc 4401   omcom 4627   ` cfv 5259   Isom wiso 5260  (class class class)co 5925   ~~ cen 6806   Fincfn 6808   0cc0 7896   1c1 7897   < clt 8078   <_ cle 8079   ZZcz 9343   QQcq 9710   ...cfz 10100  ♯chash 10884

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-ihash 10885

This theorem is referenced by:  summodclem2  11564  zsumdc  11566  prodmodclem2  11759  zproddc  11761