Theorem fin23lem22 9149

Description: Lemma for fin23 9211 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 9150) between an infinite subset of om and om itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

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

Assertion

Ref	Expression
fin23lem22	|- ( ( S C_ om /\ -. S e. Fin ) -> C : om -1-1-onto-> S )

Distinct variable group:   i, j, S

Allowed substitution hints:   C( i, j)

Proof of Theorem fin23lem22

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem22.b	. 2 |- C = ( i e. om |-> ( iota_ j e. S ( j i^i S ) ~~ i ) )
2		fin23lem23 9148	. . 3 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ i e. om ) -> E! j e. S ( j i^i S ) ~~ i )
3		riotacl 6625	. . 3 |- ( E! j e. S ( j i^i S ) ~~ i -> ( iota_ j e. S ( j i^i S ) ~~ i ) e. S )
4	2, 3	syl 17	. 2 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ i e. om ) -> ( iota_ j e. S ( j i^i S ) ~~ i ) e. S )
5		simpll 790	. . . 4 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ a e. S ) -> S C_ om )
6		simpr 477	. . . 4 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ a e. S ) -> a e. S )
7	5, 6	sseldd 3604	. . 3 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ a e. S ) -> a e. om )
8		nnfi 8153	. . 3 |- ( a e. om -> a e. Fin )
9		infi 8184	. . 3 |- ( a e. Fin -> ( a i^i S ) e. Fin )
10		ficardom 8787	. . 3 |- ( ( a i^i S ) e. Fin -> ( card ` ( a i^i S ) ) e. om )
11	7, 8, 9, 10	4syl 19	. 2 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ a e. S ) -> ( card ` ( a i^i S ) ) e. om )
12		cardnn 8789	. . . . . . 7 |- ( i e. om -> ( card ` i ) = i )
13	12	eqcomd 2628	. . . . . 6 |- ( i e. om -> i = ( card ` i ) )
14	13	eqeq1d 2624	. . . . 5 |- ( i e. om -> ( i = ( card ` ( a i^i S ) ) <-> ( card ` i ) = ( card ` ( a i^i S ) ) ) )
15		eqcom 2629	. . . . 5 |- ( ( card ` i ) = ( card ` ( a i^i S ) ) <-> ( card ` ( a i^i S ) ) = ( card ` i ) )
16	14, 15	syl6bb 276	. . . 4 |- ( i e. om -> ( i = ( card ` ( a i^i S ) ) <-> ( card ` ( a i^i S ) ) = ( card ` i ) ) )
17	16	ad2antrl 764	. . 3 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> ( i = ( card ` ( a i^i S ) ) <-> ( card ` ( a i^i S ) ) = ( card ` i ) ) )
18		simpll 790	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> S C_ om )
19		simprr 796	. . . . . . 7 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> a e. S )
20	18, 19	sseldd 3604	. . . . . 6 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> a e. om )
21		nnon 7071	. . . . . 6 |- ( a e. om -> a e. On )
22		onenon 8775	. . . . . 6 |- ( a e. On -> a e. dom card )
23	20, 21, 22	3syl 18	. . . . 5 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> a e. dom card )
24		inss1 3833	. . . . 5 |- ( a i^i S ) C_ a
25		ssnum 8862	. . . . 5 |- ( ( a e. dom card /\ ( a i^i S ) C_ a ) -> ( a i^i S ) e. dom card )
26	23, 24, 25	sylancl 694	. . . 4 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> ( a i^i S ) e. dom card )
27		nnon 7071	. . . . . 6 |- ( i e. om -> i e. On )
28	27	ad2antrl 764	. . . . 5 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> i e. On )
29		onenon 8775	. . . . 5 |- ( i e. On -> i e. dom card )
30	28, 29	syl 17	. . . 4 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> i e. dom card )
31		carden2 8813	. . . 4 |- ( ( ( a i^i S ) e. dom card /\ i e. dom card ) -> ( ( card ` ( a i^i S ) ) = ( card ` i ) <-> ( a i^i S ) ~~ i ) )
32	26, 30, 31	syl2anc 693	. . 3 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> ( ( card ` ( a i^i S ) ) = ( card ` i ) <-> ( a i^i S ) ~~ i ) )
33	2	adantrr 753	. . . . 5 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> E! j e. S ( j i^i S ) ~~ i )
34		ineq1 3807	. . . . . . 7 |- ( j = a -> ( j i^i S ) = ( a i^i S ) )
35	34	breq1d 4663	. . . . . 6 |- ( j = a -> ( ( j i^i S ) ~~ i <-> ( a i^i S ) ~~ i ) )
36	35	riota2 6633	. . . . 5 |- ( ( a e. S /\ E! j e. S ( j i^i S ) ~~ i ) -> ( ( a i^i S ) ~~ i <-> ( iota_ j e. S ( j i^i S ) ~~ i ) = a ) )
37	19, 33, 36	syl2anc 693	. . . 4 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> ( ( a i^i S ) ~~ i <-> ( iota_ j e. S ( j i^i S ) ~~ i ) = a ) )
38		eqcom 2629	. . . 4 |- ( ( iota_ j e. S ( j i^i S ) ~~ i ) = a <-> a = ( iota_ j e. S ( j i^i S ) ~~ i ) )
39	37, 38	syl6bb 276	. . 3 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> ( ( a i^i S ) ~~ i <-> a = ( iota_ j e. S ( j i^i S ) ~~ i ) ) )
40	17, 32, 39	3bitrd 294	. 2 |- ( ( ( S C_ om /\ -. S e. Fin ) /\ ( i e. om /\ a e. S ) ) -> ( i = ( card ` ( a i^i S ) ) <-> a = ( iota_ j e. S ( j i^i S ) ~~ i ) ) )
41	1, 4, 11, 40	f1o2d 6887	1 |- ( ( S C_ om /\ -. S e. Fin ) -> C : om -1-1-onto-> S )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!wreu 2914   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Oncon0 5723   -1-1-onto->wf1o 5887   ` cfv 5888   iota_crio 6610   omcom 7065   ~~ cen 7952   Fincfn 7955   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin23lem27  9150  fin23lem28  9162  fin23lem30  9164  isf32lem6  9180  isf32lem7  9181  isf32lem8  9182