Theorem zfz1iso 10988

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 6854	. . . 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 4049	. . . . . . . . 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 1847	. . . . . 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 4049	. . . . . . . . 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 1847	. . . . . 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 4049	. . . . . . . . 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 1847	. . . . . 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 4049	. . . . . . . . 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 1847	. . . . . 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 5888	. . . . . . . . . 10 |- (/) Isom < , < ( (/) , (/) )
22		en0 6889	. . . . . . . . . . . . . . . . 17 |- ( x ~~ (/) <-> x = (/) )
23	22	biimpi 120	. . . . . . . . . . . . . . . 16 |- ( x ~~ (/) -> x = (/) )
24	23	fveq2d 5582	. . . . . . . . . . . . . . 15 |- ( x ~~ (/) -> (♯ ` x ) = (♯ ` (/) ) )
25		hash0 10943	. . . . . . . . . . . . . . 15 |- (♯ ` (/) ) = 0
26	24, 25	eqtrdi 2254	. . . . . . . . . . . . . 14 |- ( x ~~ (/) -> (♯ ` x ) = 0 )
27	26	oveq2d 5962	. . . . . . . . . . . . 13 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = ( 1 ... 0 ) )
28		fz10 10170	. . . . . . . . . . . . 13 |- ( 1 ... 0 ) = (/)
29	27, 28	eqtrdi 2254	. . . . . . . . . . . 12 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = (/) )
30		isoeq4 5875	. . . . . . . . . . . 12 |- ( ( 1 ... (♯ ` x ) ) = (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
31	29, 30	syl 14	. . . . . . . . . . 11 |- ( x ~~ (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
32		isoeq5 5876	. . . . . . . . . . . 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 4172	. . . . . . . . . 10 |- (/) e. _V
37		isoeq1 5872	. . . . . . . . . 10 |- ( f = (/) -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
38	36, 37	spcev 2868	. . . . . . . . 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 1472	. . . . . 6 |- A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
42		sseq1 3216	. . . . . . . . . . 11 |- ( a = x -> ( a C_ ZZ <-> x C_ ZZ ) )
43		eleq1 2268	. . . . . . . . . . 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 4048	. . . . . . . . . 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 5578	. . . . . . . . . . . . 13 |- ( a = x -> (♯ ` a ) = (♯ ` x ) )
48	47	oveq2d 5962	. . . . . . . . . . . 12 |- ( a = x -> ( 1 ... (♯ ` a ) ) = ( 1 ... (♯ ` x ) ) )
49		isoeq4 5875	. . . . . . . . . . . 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 5876	. . . . . . . . . . 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 1848	. . . . . . . . 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 1941	. . . . . . 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 9750	. . . . . . . . . . . . 13 |- ZZ C_ QQ
58	56, 57	sstrdi 3205	. . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . 16 |- k e. _V
61		nsuceq0g 4466	. . . . . . . . . . . . . . . 16 |- ( k e. _V -> suc k =/= (/) )
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 |- suc k =/= (/)
63	62	neii 2378	. . . . . . . . . . . . . 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 6877	. . . . . . . . . . . . . . . . 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 4071	. . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . 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 666	. . . . . . . . . . . . 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 2383	. . . . . . . . . . . 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 10974	. . . . . . . . . . . 12 |- ( ( x C_ QQ /\ x e. Fin /\ x =/= (/) ) -> E. m e. x A. z e. x z <_ m )
74	58, 59, 72, 73	syl3anc 1250	. . . . . . . . . . 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 10987	. . . . . . . . . . 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 2628	. . . . . . . . . 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 1897	. . . . . . . 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 4649	. . . . 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 3216	. . . . . . . 8 |- ( x = A -> ( x C_ ZZ <-> A C_ ZZ ) )
92		eleq1 2268	. . . . . . . 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 4048	. . . . . . 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 5578	. . . . . . . . . 10 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
97	96	oveq2d 5962	. . . . . . . . 9 |- ( x = A -> ( 1 ... (♯ ` x ) ) = ( 1 ... (♯ ` A ) ) )
98		isoeq4 5875	. . . . . . . . 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 5876	. . . . . . . 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 1848	. . . . . 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 2860	. . . 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 2628	1 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   =/= wne 2376   A.wral 2484   E.wrex 2485   _Vcvv 2772   C_ wss 3166   (/)c0 3460   class class class wbr 4045   suc csuc 4413   omcom 4639   ` cfv 5272   Isom wiso 5273  (class class class)co 5946   ~~ cen 6827   Fincfn 6829   0cc0 7927   1c1 7928   < clt 8109   <_ cle 8110   ZZcz 9374   QQcq 9742   ...cfz 10132  ♯chash 10922

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-ihash 10923

This theorem is referenced by:  summodclem2  11726  zsumdc  11728  prodmodclem2  11921  zproddc  11923