Theorem fin23lem27 8493

Description: The mapping constructed in fin23lem22 8492 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 6484	. . . 4 |- Ord om
2		ordwe 4728	. . . 4 |- ( Ord om -> _E We om )
3		weso 4707	. . . 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 4654	. . . . 5 |- ( _E Or om -> _E Po om )
7	4, 6	ax-mp 5	. . . 4 |- _E Po om
8		poss 4639	. . . 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 462	. 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 8492	. . 3 |- ( ( S C_ om /\ -. S e. Fin ) -> C : om -1-1-onto-> S )
13		f1ofo 5645	. . 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 8156	. . . . . . . 8 |- ( ( a e. om /\ b e. om ) -> ( a e. b <-> a ~< b ) )
16	15	adantl 463	. . . . . . 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 8491	. . . . . . . . . . . . 13 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ a e. om ) -> E! j e. S ( j i^i S ) ~~ a )
19	18	adantrr 711	. . . . . . . . . . . 12 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> E! j e. S ( j i^i S ) ~~ a )
20		ineq1 3542	. . . . . . . . . . . . . 14 |- ( j = i -> ( j i^i S ) = ( i i^i S ) )
21	20	breq1d 4299	. . . . . . . . . . . . 13 |- ( j = i -> ( ( j i^i S ) ~~ a <-> ( i i^i S ) ~~ a ) )
22	21	cbvreuv 2947	. . . . . . . . . . . 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 1678	. . . . . . . . . . . 12 |- F/ i ( ( iota_ j e. S ( j i^i S ) ~~ a ) i^i S ) ~~ a
25	21	cbvriotav 6061	. . . . . . . . . . . 12 |- ( iota_ j e. S ( j i^i S ) ~~ a ) = ( iota_ i e. S ( i i^i S ) ~~ a )
26		ineq1 3542	. . . . . . . . . . . . 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 4299	. . . . . . . . . . . 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 6074	. . . . . . . . . . 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 460	. . . . . . . . 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 711	. . . . . . . 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 751	. . . . . . . . 9 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( ( a e. om /\ b e. om ) /\ a ~< b ) ) -> a ~< b )
33		fin23lem23 8491	. . . . . . . . . . . . . . 15 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ b e. om ) -> E! j e. S ( j i^i S ) ~~ b )
34	33	adantrl 710	. . . . . . . . . . . . . 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 4299	. . . . . . . . . . . . . . 15 |- ( j = i -> ( ( j i^i S ) ~~ b <-> ( i i^i S ) ~~ b ) )
36	35	cbvreuv 2947	. . . . . . . . . . . . . 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 1678	. . . . . . . . . . . . . 14 |- F/ i ( ( iota_ j e. S ( j i^i S ) ~~ b ) i^i S ) ~~ b
39	35	cbvriotav 6061	. . . . . . . . . . . . . 14 |- ( iota_ j e. S ( j i^i S ) ~~ b ) = ( iota_ i e. S ( i i^i S ) ~~ b )
40		ineq1 3542	. . . . . . . . . . . . . . 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 4299	. . . . . . . . . . . . . 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 6074	. . . . . . . . . . . . 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 460	. . . . . . . . . . 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 7356	. . . . . . . . . 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 711	. . . . . . . . 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 7441	. . . . . . . . 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 656	. . . . . . . 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 7443	. . . . . . . 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 656	. . . . . . 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 612	. . . . . 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 748	. . . . . . . . 9 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> S C_ om )
53		omsson 6479	. . . . . . . . 9 |- om C_ On
54	52, 53	syl6ss 3365	. . . . . . . 8 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> S C_ On )
55	29	simpld 456	. . . . . . . 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 3354	. . . . . . 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 456	. . . . . . . 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 3354	. . . . . . 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 7456	. . . . . . . 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 1184	. . . . . . 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 656	. . . . . 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 750	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> a e. om )
64		breq2 4293	. . . . . . . . 9 |- ( i = a -> ( ( j i^i S ) ~~ i <-> ( j i^i S ) ~~ a ) )
65	64	riotabidv 6051	. . . . . . . 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 5769	. . . . . . 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 656	. . . . . 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 751	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( a e. om /\ b e. om ) ) -> b e. om )
69		breq2 4293	. . . . . . . . 9 |- ( i = b -> ( ( j i^i S ) ~~ i <-> ( j i^i S ) ~~ b ) )
70	69	riotabidv 6051	. . . . . . . 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 5769	. . . . . . 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 656	. . . . . 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 2509	. . . . 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 4631	. . . 4 |- ( a _E b <-> a e. b )
76		fvex 5698	. . . . 5 |- ( C ` b ) e. _V
77	76	epelc 4630	. . . 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 2806	. 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 6016	. 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 1214	1 |- ( ( S C_ om /\ -. S e. Fin ) -> C Isom _E , _E ( om , S ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   A.wral 2713   E!wreu 2715   i^i cin 3324   C_ wss 3325   class class class wbr 4289   e. cmpt 4347   _E cep 4626   Po wpo 4635   Or wor 4636   We wwe 4674   Ord word 4714   Oncon0 4715   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415   Isom wiso 5416   iota_crio 6048   omcom 6475   ~~ cen 7303   ~< csdm 7305   Fincfn 7306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-om 6476  df-recs 6828  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105

This theorem is referenced by: (None)