Theorem zfz1iso 10754

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 6727	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 119	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	adantl 275	. 2 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. n e. om A ~~ n )
4		simprlr 528	. . . 4 |- ( ( n e. om /\ ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) -> A e. Fin )
5		breq2 3986	. . . . . . . . 9 |- ( w = (/) -> ( x ~~ w <-> x ~~ (/) ) )
6	5	anbi2d 460	. . . . . . . 8 |- ( w = (/) -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) ) )
7	6	imbi1d 230	. . . . . . 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 1812	. . . . . 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 3986	. . . . . . . . 9 |- ( w = k -> ( x ~~ w <-> x ~~ k ) )
10	9	anbi2d 460	. . . . . . . 8 |- ( w = k -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) ) )
11	10	imbi1d 230	. . . . . . 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 1812	. . . . . 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 3986	. . . . . . . . 9 |- ( w = suc k -> ( x ~~ w <-> x ~~ suc k ) )
14	13	anbi2d 460	. . . . . . . 8 |- ( w = suc k -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ suc k ) ) )
15	14	imbi1d 230	. . . . . . 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 1812	. . . . . 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 3986	. . . . . . . . 9 |- ( w = n -> ( x ~~ w <-> x ~~ n ) )
18	17	anbi2d 460	. . . . . . . 8 |- ( w = n -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ w ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) ) )
19	18	imbi1d 230	. . . . . . 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 1812	. . . . . 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 5785	. . . . . . . . . 10 |- (/) Isom < , < ( (/) , (/) )
22		en0 6761	. . . . . . . . . . . . . . . . 17 |- ( x ~~ (/) <-> x = (/) )
23	22	biimpi 119	. . . . . . . . . . . . . . . 16 |- ( x ~~ (/) -> x = (/) )
24	23	fveq2d 5490	. . . . . . . . . . . . . . 15 |- ( x ~~ (/) -> (♯ ` x ) = (♯ ` (/) ) )
25		hash0 10710	. . . . . . . . . . . . . . 15 |- (♯ ` (/) ) = 0
26	24, 25	eqtrdi 2215	. . . . . . . . . . . . . 14 |- ( x ~~ (/) -> (♯ ` x ) = 0 )
27	26	oveq2d 5858	. . . . . . . . . . . . 13 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = ( 1 ... 0 ) )
28		fz10 9981	. . . . . . . . . . . . 13 |- ( 1 ... 0 ) = (/)
29	27, 28	eqtrdi 2215	. . . . . . . . . . . 12 |- ( x ~~ (/) -> ( 1 ... (♯ ` x ) ) = (/) )
30		isoeq4 5772	. . . . . . . . . . . 12 |- ( ( 1 ... (♯ ` x ) ) = (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
31	29, 30	syl 14	. . . . . . . . . . 11 |- ( x ~~ (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , x ) ) )
32		isoeq5 5773	. . . . . . . . . . . 12 |- ( x = (/) -> ( (/) Isom < , < ( (/) , x ) <-> (/) Isom < , < ( (/) , (/) ) ) )
33	23, 32	syl 14	. . . . . . . . . . 11 |- ( x ~~ (/) -> ( (/) Isom < , < ( (/) , x ) <-> (/) Isom < , < ( (/) , (/) ) ) )
34	31, 33	bitrd 187	. . . . . . . . . 10 |- ( x ~~ (/) -> ( (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( (/) , (/) ) ) )
35	21, 34	mpbiri 167	. . . . . . . . 9 |- ( x ~~ (/) -> (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
36		0ex 4109	. . . . . . . . . 10 |- (/) e. _V
37		isoeq1 5769	. . . . . . . . . 10 |- ( f = (/) -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> (/) Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
38	36, 37	spcev 2821	. . . . . . . . 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 275	. . . . . . 7 |- ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
41	40	ax-gen 1437	. . . . . 6 |- A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ (/) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) )
42		sseq1 3165	. . . . . . . . . . 11 |- ( a = x -> ( a C_ ZZ <-> x C_ ZZ ) )
43		eleq1 2229	. . . . . . . . . . 11 |- ( a = x -> ( a e. Fin <-> x e. Fin ) )
44	42, 43	anbi12d 465	. . . . . . . . . 10 |- ( a = x -> ( ( a C_ ZZ /\ a e. Fin ) <-> ( x C_ ZZ /\ x e. Fin ) ) )
45		breq1 3985	. . . . . . . . . 10 |- ( a = x -> ( a ~~ k <-> x ~~ k ) )
46	44, 45	anbi12d 465	. . . . . . . . 9 |- ( a = x -> ( ( ( a C_ ZZ /\ a e. Fin ) /\ a ~~ k ) <-> ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ k ) ) )
47		fveq2 5486	. . . . . . . . . . . . 13 |- ( a = x -> (♯ ` a ) = (♯ ` x ) )
48	47	oveq2d 5858	. . . . . . . . . . . 12 |- ( a = x -> ( 1 ... (♯ ` a ) ) = ( 1 ... (♯ ` x ) ) )
49		isoeq4 5772	. . . . . . . . . . . 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 5773	. . . . . . . . . . 11 |- ( a = x -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , a ) <-> f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
52	50, 51	bitrd 187	. . . . . . . . . 10 |- ( a = x -> ( f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) <-> f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
53	52	exbidv 1813	. . . . . . . . 9 |- ( a = x -> ( E. f f Isom < , < ( ( 1 ... (♯ ` a ) ) , a ) <-> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
54	46, 53	imbi12d 233	. . . . . . . 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 1905	. . . . . . 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 527	. . . . . . . . . . . . 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 9565	. . . . . . . . . . . . 13 |- ZZ C_ QQ
58	56, 57	sstrdi 3154	. . . . . . . . . . . 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 528	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 |- k e. _V
61		nsuceq0g 4396	. . . . . . . . . . . . . . . 16 |- ( k e. _V -> suc k =/= (/) )
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 |- suc k =/= (/)
63	62	neii 2338	. . . . . . . . . . . . . 14 |- -. suc k = (/)
64		simplrr 526	. . . . . . . . . . . . . . . . . 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 6749	. . . . . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . . 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 6761	. . . . . . . . . . . . . . . 16 |- ( suc k ~~ (/) <-> suc k = (/) )
69	67, 68	sylib 121	. . . . . . . . . . . . . . 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 114	. . . . . . . . . . . . . 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 654	. . . . . . . . . . . . 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 2343	. . . . . . . . . . . 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 10740	. . . . . . . . . . . 12 |- ( ( x C_ QQ /\ x e. Fin /\ x =/= (/) ) -> E. m e. x A. z e. x z <_ m )
74	58, 59, 72, 73	syl3anc 1228	. . . . . . . . . . 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 523	. . . . . . . . . . . 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 524	. . . . . . . . . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . 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 526	. . . . . . . . . . . 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 521	. . . . . . . . . . . 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 522	. . . . . . . . . . . 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 10753	. . . . . . . . . . 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 2588	. . . . . . . . . 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 114	. . . . . . . . 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 1862	. . . . . . . 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 114	. . . . . . 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	syl5bir 152	. . . . . 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 4577	. . . . 5 |- ( n e. om -> A. x ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) -> E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) ) )
89	88	adantr 274	. . . 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 109	. . . 4 |- ( ( n e. om /\ ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) -> ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) )
91		sseq1 3165	. . . . . . . 8 |- ( x = A -> ( x C_ ZZ <-> A C_ ZZ ) )
92		eleq1 2229	. . . . . . . 8 |- ( x = A -> ( x e. Fin <-> A e. Fin ) )
93	91, 92	anbi12d 465	. . . . . . 7 |- ( x = A -> ( ( x C_ ZZ /\ x e. Fin ) <-> ( A C_ ZZ /\ A e. Fin ) ) )
94		breq1 3985	. . . . . . 7 |- ( x = A -> ( x ~~ n <-> A ~~ n ) )
95	93, 94	anbi12d 465	. . . . . 6 |- ( x = A -> ( ( ( x C_ ZZ /\ x e. Fin ) /\ x ~~ n ) <-> ( ( A C_ ZZ /\ A e. Fin ) /\ A ~~ n ) ) )
96		fveq2 5486	. . . . . . . . . 10 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
97	96	oveq2d 5858	. . . . . . . . 9 |- ( x = A -> ( 1 ... (♯ ` x ) ) = ( 1 ... (♯ ` A ) ) )
98		isoeq4 5772	. . . . . . . . 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 5773	. . . . . . . 8 |- ( x = A -> ( f Isom < , < ( ( 1 ... (♯ ` A ) ) , x ) <-> f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) )
101	99, 100	bitrd 187	. . . . . . 7 |- ( x = A -> ( f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) )
102	101	exbidv 1813	. . . . . 6 |- ( x = A -> ( E. f f Isom < , < ( ( 1 ... (♯ ` x ) ) , x ) <-> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) )
103	95, 102	imbi12d 233	. . . . 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 2813	. . . 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 555	. 2 |- ( ( ( A C_ ZZ /\ A e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
107	3, 106	rexlimddv 2588	1 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   =/= wne 2336   A.wral 2444   E.wrex 2445   _Vcvv 2726   C_ wss 3116   (/)c0 3409   class class class wbr 3982   suc csuc 4343   omcom 4567   ` cfv 5188   Isom wiso 5189  (class class class)co 5842   ~~ cen 6704   Fincfn 6706   0cc0 7753   1c1 7754   < clt 7933   <_ cle 7934   ZZcz 9191   QQcq 9557   ...cfz 9944  ♯chash 10688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-ihash 10689

This theorem is referenced by:  summodclem2  11323  zsumdc  11325  prodmodclem2  11518  zproddc  11520