Theorem fin23lem27 8509

Description: The mapping constructed in fin23lem22 8508 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem22.b	|- C = ( i e. om |-> ( iota_ j e. S ( j i^i S ) ~~ i ) )

Assertion

Ref	Expression
fin23lem27	|- ( ( S C_ om /\ -. S e. Fin ) -> C Isom _E , _E ( om , S ) )

Distinct variable group:   i, j, S

Allowed substitution hints:   C( i, j)

Proof of Theorem fin23lem27

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordom 6497	. . . 4 |- Ord om
2		ordwe 4744	. . . 4 |- ( Ord om -> _E We om )
3		weso 4723	. . . 4 |- ( _E We om -> _E Or om )
4	1, 2, 3	mp2b 10	. . 3 |- _E Or om
5	4	a1i 11	. 2 |- ( ( S C_ om /\ -. S e. Fin ) -> _E Or om )
6		sopo 4670	. . . . 5 |- ( _E Or om -> _E Po om )
7	4, 6	ax-mp 5	. . . 4 |- _E Po om
8		poss 4655	. . . 4 |- ( S C_ om -> ( _E Po om -> _E Po S ) )
9	7, 8	mpi 17	. . 3 |- ( S C_ om -> _E Po S )
10	9	adantr 465	. 2 |- ( ( S C_ om /\ -. S e. Fin ) -> _E Po S )
11		fin23lem22.b	. . . 4 |- C = ( i e. om |-> ( iota_ j e. S ( j i^i S ) ~~ i ) )
12	11	fin23lem22 8508	. . 3 |- ( ( S C_ om /\ -. S e. Fin ) -> C : om -1-1-onto-> S )
13		f1ofo 5660	. . 3 |- ( C : om -1-1-onto-> S -> C : om -onto-> S )
14	12, 13	syl 16	. 2 |- ( ( S C_ om /\ -. S e. Fin ) -> C : om -onto-> S )
15		nnsdomel 8172	. . . . . . . 8 |- ( ( a e. om /\ b e. om ) -> ( a e. b <-> a ~< b ) )
16	15	adantl 466	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( a e. b <-> a ~< b ) )
17	16	biimpd 207	. . . . . 6 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( a e. b -> a ~< b ) )
18		fin23lem23 8507	. . . . . . . . . . . . 13 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ a e. om ) -> E! j e. S ( j i^i S ) ~~ a )
19	18	adantrr 716	. . . . . . . . . . . 12 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> E! j e. S ( j i^i S ) ~~ a )
20		ineq1 3557	. . . . . . . . . . . . . 14 |- ( j = i -> ( j i^i S ) = ( i i^i S ) )
21	20	breq1d 4314	. . . . . . . . . . . . 13 |- ( j = i -> ( ( j i^i S ) ~~ a <-> ( i i^i S ) ~~ a ) )
22	21	cbvreuv 2961	. . . . . . . . . . . 12 |- ( E! j e. S ( j i^i S ) ~~ a <-> E! i e. S ( i i^i S ) ~~ a )
23	19, 22	sylib 196	. . . . . . . . . . 11 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> E! i e. S ( i i^i S ) ~~ a )
24		nfv 1673	. . . . . . . . . . . 12 |- F/ i ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a
25	21	cbvriotav 6075	. . . . . . . . . . . 12 |- ( iota_ j e. S ( j i^i S ) ~~ a ) = ( iota_ i e. S ( i i^i S ) ~~ a )
26		ineq1 3557	. . . . . . . . . . . . 13 |- ( i = ( iota_ j e. S ( j i^i S ) ~~ a ) -> ( i i^i S ) = ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) )
27	26	breq1d 4314	. . . . . . . . . . . 12 |- ( i = ( iota_ j e. S ( j i^i S ) ~~ a ) -> ( ( i i^i S ) ~~ a <-> ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a ) )
28	24, 25, 27	riotaprop 6088	. . . . . . . . . . 11 |- ( E! i e. S ( i i^i S ) ~~ a -> ( ( iota_ j e. S ( j i^i S ) ~~ a ) e. S /\ ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a ) )
29	23, 28	syl 16	. . . . . . . . . 10 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( ( iota_ j e. S ( j i^i S ) ~~ a ) e. S /\ ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a ) )
30	29	simprd 463	. . . . . . . . 9 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a )
31	30	adantrr 716	. . . . . . . 8 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( ( a e. om /\ b e. om ) /\ a ~< b ) ) -> ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a )
32		simprr 756	. . . . . . . . 9 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( ( a e. om /\ b e. om ) /\ a ~< b ) ) -> a ~< b )
33		fin23lem23 8507	. . . . . . . . . . . . . . 15 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ b e. om ) -> E! j e. S ( j i^i S ) ~~ b )
34	33	adantrl 715	. . . . . . . . . . . . . 14 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> E! j e. S ( j i^i S ) ~~ b )
35	20	breq1d 4314	. . . . . . . . . . . . . . 15 |- ( j = i -> ( ( j i^i S ) ~~ b <-> ( i i^i S ) ~~ b ) )
36	35	cbvreuv 2961	. . . . . . . . . . . . . 14 |- ( E! j e. S ( j i^i S ) ~~ b <-> E! i e. S ( i i^i S ) ~~ b )
37	34, 36	sylib 196	. . . . . . . . . . . . 13 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> E! i e. S ( i i^i S ) ~~ b )
38		nfv 1673	. . . . . . . . . . . . . 14 |- F/ i ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ~~ b
39	35	cbvriotav 6075	. . . . . . . . . . . . . 14 |- ( iota_ j e. S ( j i^i S ) ~~ b ) = ( iota_ i e. S ( i i^i S ) ~~ b )
40		ineq1 3557	. . . . . . . . . . . . . . 15 |- ( i = ( iota_ j e. S ( j i^i S ) ~~ b ) -> ( i i^i S ) = ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) )
41	40	breq1d 4314	. . . . . . . . . . . . . 14 |- ( i = ( iota_ j e. S ( j i^i S ) ~~ b ) -> ( ( i i^i S ) ~~ b <-> ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ~~ b ) )
42	38, 39, 41	riotaprop 6088	. . . . . . . . . . . . 13 |- ( E! i e. S ( i i^i S ) ~~ b -> ( ( iota_ j e. S ( j i^i S ) ~~ b ) e. S /\ ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ~~ b ) )
43	37, 42	syl 16	. . . . . . . . . . . 12 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( ( iota_ j e. S ( j i^i S ) ~~ b ) e. S /\ ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ~~ b ) )
44	43	simprd 463	. . . . . . . . . . 11 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ~~ b )
45	44	ensymd 7372	. . . . . . . . . 10 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> b ~~ ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) )
46	45	adantrr 716	. . . . . . . . 9 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( ( a e. om /\ b e. om ) /\ a ~< b ) ) -> b ~~ ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) )
47		sdomentr 7457	. . . . . . . . 9 |- ( ( a ~< b /\ b ~~ ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ) -> a ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) )
48	32, 46, 47	syl2anc 661	. . . . . . . 8 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( ( a e. om /\ b e. om ) /\ a ~< b ) ) -> a ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) )
49		ensdomtr 7459	. . . . . . . 8 |- ( ( ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a /\ a ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ) -> ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) )
50	31, 48, 49	syl2anc 661	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( ( a e. om /\ b e. om ) /\ a ~< b ) ) -> ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) )
51	50	expr 615	. . . . . 6 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( a ~< b -> ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ) )
52		simpll 753	. . . . . . . . 9 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> S C_ om )
53		omsson 6492	. . . . . . . . 9 |- om C_ On
54	52, 53	syl6ss 3380	. . . . . . . 8 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> S C_ On )
55	29	simpld 459	. . . . . . . 8 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( iota_ j e. S ( j i^i S ) ~~ a ) e. S )
56	54, 55	sseldd 3369	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( iota_ j e. S ( j i^i S ) ~~ a ) e. On )
57	43	simpld 459	. . . . . . . 8 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( iota_ j e. S ( j i^i S ) ~~ b ) e. S )
58	54, 57	sseldd 3369	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( iota_ j e. S ( j i^i S ) ~~ b ) e. On )
59		onsdominel 7472	. . . . . . . 8 |- ( ( ( iota_ j e. S ( j i^i S ) ~~ a ) e. On /\ ( iota_ j e. S ( j i^i S ) ~~ b ) e. On /\ ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ) -> ( iota_ j e. S ( j i^i S ) ~~ a ) e. ( iota_ j e. S ( j i^i S ) ~~ b ) )
60	59	3expia 1189	. . . . . . 7 |- ( ( ( iota_ j e. S ( j i^i S ) ~~ a ) e. On /\ ( iota_ j e. S ( j i^i S ) ~~ b ) e. On ) -> ( ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) -> ( iota_ j e. S ( j i^i S ) ~~ a ) e. ( iota_ j e. S ( j i^i S ) ~~ b ) ) )
61	56, 58, 60	syl2anc 661	. . . . . 6 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~< ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) -> ( iota_ j e. S ( j i^i S ) ~~ a ) e. ( iota_ j e. S ( j i^i S ) ~~ b ) ) )
62	17, 51, 61	3syld 55	. . . . 5 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( a e. b -> ( iota_ j e. S ( j i^i S ) ~~ a ) e. ( iota_ j e. S ( j i^i S ) ~~ b ) ) )
63		simprl 755	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> a e. om )
64		breq2 4308	. . . . . . . . 9 |- ( i = a -> ( ( j i^i S ) ~~ i <-> ( j i^i S ) ~~ a ) )
65	64	riotabidv 6066	. . . . . . . 8 |- ( i = a -> ( iota_ j e. S ( j i^i S ) ~~ i ) = ( iota_ j e. S ( j i^i S ) ~~ a ) )
66	65, 11	fvmptg 5784	. . . . . . 7 |- ( ( a e. om /\ ( iota_ j e. S ( j i^i S ) ~~ a ) e. S ) -> ( C ` a ) = ( iota_ j e. S ( j i^i S ) ~~ a ) )
67	63, 55, 66	syl2anc 661	. . . . . 6 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( C ` a ) = ( iota_ j e. S ( j i^i S ) ~~ a ) )
68		simprr 756	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> b e. om )
69		breq2 4308	. . . . . . . . 9 |- ( i = b -> ( ( j i^i S ) ~~ i <-> ( j i^i S ) ~~ b ) )
70	69	riotabidv 6066	. . . . . . . 8 |- ( i = b -> ( iota_ j e. S ( j i^i S ) ~~ i ) = ( iota_ j e. S ( j i^i S ) ~~ b ) )
71	70, 11	fvmptg 5784	. . . . . . 7 |- ( ( b e. om /\ ( iota_ j e. S ( j i^i S ) ~~ b ) e. S ) -> ( C ` b ) = ( iota_ j e. S ( j i^i S ) ~~ b ) )
72	68, 57, 71	syl2anc 661	. . . . . 6 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( C ` b ) = ( iota_ j e. S ( j i^i S ) ~~ b ) )
73	67, 72	eleq12d 2511	. . . . 5 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( ( C ` a ) e. ( C ` b ) <-> ( iota_ j e. S ( j i^i S ) ~~ a ) e. ( iota_ j e. S ( j i^i S ) ~~ b ) ) )
74	62, 73	sylibrd 234	. . . 4 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( a e. b -> ( C ` a ) e. ( C ` b ) ) )
75		epel 4647	. . . 4 |- ( a _E b <-> a e. b )
76		fvex 5713	. . . . 5 |- ( C ` b ) e. _V
77	76	epelc 4646	. . . 4 |- ( ( C ` a ) _E ( C ` b ) <-> ( C ` a ) e. ( C ` b ) )
78	74, 75, 77	3imtr4g 270	. . 3 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> ( a _E b -> ( C ` a ) _E ( C ` b ) ) )
79	78	ralrimivva 2820	. 2 |- ( ( S C_ om /\ -. S e. Fin ) -> A. a e. om A. b e. om ( a _E b -> ( C ` a ) _E ( C ` b ) ) )
80		soisoi 6031	. 2 |- ( ( ( _E Or om /\ _E Po S ) /\ ( C : om -onto-> S /\ A. a e. om A. b e. om ( a _E b -> ( C ` a ) _E ( C ` b ) ) ) ) -> C Isom _E , _E ( om , S ) )
81	5, 10, 14, 79, 80	syl22anc 1219	1 |- ( ( S C_ om /\ -. S e. Fin ) -> C Isom _E , _E ( om , S ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2727   E!wreu 2729   i^i cin 3339   C_ wss 3340   class class class wbr 4304   e. cmpt 4362   _E cep 4642   Po wpo 4651   Or wor 4652   We wwe 4690   Ord word 4730   Oncon0 4731   -onto->wfo 5428   -1-1-onto->wf1o 5429   ` cfv 5430   Isom wiso 5431   iota_crio 6063   omcom 6488   ~~ cen 7319   ~< csdm 7321   Fincfn 7322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-om 6489  df-recs 6844  df-1o 6932  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121

This theorem is referenced by: (None)