Theorem zfz1iso 10823

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 6763	. . . 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 4009	. . . . . . . . 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 1824	. . . . . 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 4009	. . . . . . . . 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 1824	. . . . . 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 4009	. . . . . . . . 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 1824	. . . . . 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 4009	. . . . . . . . 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 1824	. . . . . 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 5820	. . . . . . . . . 10 |- (/) Isom < , < ( (/) , (/) )
22		en0 6797	. . . . . . . . . . . . . . . . 17 |- ( x ~~ (/) <-> x = (/) )
23	22	biimpi 120	. . . . . . . . . . . . . . . 16 |- ( x ~~ (/) -> x = (/) )
24	23	fveq2d 5521	. . . . . . . . . . . . . . 15 |- ( x ~~ (/) -> (♯ ` x ) = (♯ ` (/) ) )
25		hash0 10778	. . . . . . . . . . . . . . 15 |- (♯ ` (/) ) = 0
26	24, 25	eqtrdi 2226	. . . . . . . . . . . . . 14 |- ( x ~~ (/) -> (♯ ` x ) = 0 )
27	26	oveq2d 5893	. . . . . . . . . . . . 13 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = ( 1 ... 0 ) )
28		fz10 10048	. . . . . . . . . . . . 13 |- ( 1 ... 0 ) = (/)
29	27, 28	eqtrdi 2226	. . . . . . . . . . . 12 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = (/) )
30		isoeq4 5807	. . . . . . . . . . . 12 |- ( ( 1 ... (♯ ` x ) ) = (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
31	29, 30	syl 14	. . . . . . . . . . 11 |- ( x ~~ (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
32		isoeq5 5808	. . . . . . . . . . . 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 4132	. . . . . . . . . 10 |- (/) e. _V
37		isoeq1 5804	. . . . . . . . . 10 |- ( f = (/) -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
38	36, 37	spcev 2834	. . . . . . . . 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 1449	. . . . . 6 |- A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
42		sseq1 3180	. . . . . . . . . . 11 |- ( a = x -> ( a C_ ZZ <-> x C_ ZZ ) )
43		eleq1 2240	. . . . . . . . . . 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 4008	. . . . . . . . . 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 5517	. . . . . . . . . . . . 13 |- ( a = x -> (♯ ` a ) = (♯ ` x ) )
48	47	oveq2d 5893	. . . . . . . . . . . 12 |- ( a = x -> ( 1 ... (♯ ` a ) ) = ( 1 ... (♯ ` x ) ) )
49		isoeq4 5807	. . . . . . . . . . . 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 5808	. . . . . . . . . . 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 1825	. . . . . . . . 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 1917	. . . . . . 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 9629	. . . . . . . . . . . . 13 |- ZZ C_ QQ
58	56, 57	sstrdi 3169	. . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . 16 |- k e. _V
61		nsuceq0g 4420	. . . . . . . . . . . . . . . 16 |- ( k e. _V -> suc k =/= (/) )
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 |- suc k =/= (/)
63	62	neii 2349	. . . . . . . . . . . . . 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 6785	. . . . . . . . . . . . . . . . 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 4031	. . . . . . . . . . . . . . . 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 6797	. . . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . 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 2354	. . . . . . . . . . . 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 10809	. . . . . . . . . . . 12 |- ( ( x C_ QQ /\ x e. Fin /\ x =/= (/) ) -> E. m e. x A. z e. x z <_ m )
74	58, 59, 72, 73	syl3anc 1238	. . . . . . . . . . 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 10822	. . . . . . . . . . 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 2599	. . . . . . . . . 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 1874	. . . . . . . 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 4601	. . . . 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 3180	. . . . . . . 8 |- ( x = A -> ( x C_ ZZ <-> A C_ ZZ ) )
92		eleq1 2240	. . . . . . . 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 4008	. . . . . . 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 5517	. . . . . . . . . 10 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
97	96	oveq2d 5893	. . . . . . . . 9 |- ( x = A -> ( 1 ... (♯ ` x ) ) = ( 1 ... (♯ ` A ) ) )
98		isoeq4 5807	. . . . . . . . 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 5808	. . . . . . . 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 1825	. . . . . 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 2826	. . . 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 2599	1 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   =/= wne 2347   A.wral 2455   E.wrex 2456   _Vcvv 2739   C_ wss 3131   (/)c0 3424   class class class wbr 4005   suc csuc 4367   omcom 4591   ` cfv 5218   Isom wiso 5219  (class class class)co 5877   ~~ cen 6740   Fincfn 6742   0cc0 7813   1c1 7814   < clt 7994   <_ cle 7995   ZZcz 9255   QQcq 9621   ...cfz 10010  ♯chash 10757

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-ihash 10758

This theorem is referenced by:  summodclem2  11392  zsumdc  11394  prodmodclem2  11587  zproddc  11589