Theorem fin4en1 8145

Description: Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
fin4en1	|- ( A ~~ B -> ( A e. FinIV -> B e. FinIV ) )

Proof of Theorem fin4en1

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ensym 7115	. 2 |- ( A ~~ B -> B ~~ A )
2		bren 7076	. . . 4 |- ( B ~~ A <-> E. f f : B -1-1-onto-> A )
3		simpr 448	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> x C. B )
4		f1of1 5632	. . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> f : B -1-1-> A )
5		pssss 3402	. . . . . . . . . . . . . 14 |- ( x C. B -> x C_ B )
6		ssid 3327	. . . . . . . . . . . . . 14 |- B C_ B
7	5, 6	jctir 525	. . . . . . . . . . . . 13 |- ( x C. B -> ( x C_ B /\ B C_ B ) )
8		f1imapss 5971	. . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ ( x C_ B /\ B C_ B ) ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
9	4, 7, 8	syl2an 464	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
10	3, 9	mpbird 224	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. ( f " B ) )
11		imadmrn 5174	. . . . . . . . . . . . . 14 |- ( f " dom f ) = ran f
12		f1odm 5637	. . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A -> dom f = B )
13	12	imaeq2d 5162	. . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ( f " dom f ) = ( f " B ) )
14		dff1o5 5642	. . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A <-> ( f : B -1-1-> A /\ ran f = A ) )
15	14	simprbi 451	. . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ran f = A )
16	11, 13, 15	3eqtr3a 2460	. . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> ( f " B ) = A )
17	16	adantr 452	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " B ) = A )
18	17	psseq2d 3400	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> ( f " x ) C. A ) )
19	10, 18	mpbid 202	. . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. A )
20	19	adantrr 698	. . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) C. A )
21		vex 2919	. . . . . . . . . . . . . 14 |- x e. _V
22	21	f1imaen 7128	. . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ x C_ B ) -> ( f " x ) ~~ x )
23	4, 5, 22	syl2an 464	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) ~~ x )
24	23	adantrr 698	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ x )
25		simprr 734	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> x ~~ B )
26		entr 7118	. . . . . . . . . . 11 |- ( ( ( f " x ) ~~ x /\ x ~~ B ) -> ( f " x ) ~~ B )
27	24, 25, 26	syl2anc 643	. . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ B )
28		vex 2919	. . . . . . . . . . . 12 |- f e. _V
29		f1oen3g 7082	. . . . . . . . . . . 12 |- ( ( f e. _V /\ f : B -1-1-onto-> A ) -> B ~~ A )
30	28, 29	mpan 652	. . . . . . . . . . 11 |- ( f : B -1-1-onto-> A -> B ~~ A )
31	30	adantr 452	. . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> B ~~ A )
32		entr 7118	. . . . . . . . . 10 |- ( ( ( f " x ) ~~ B /\ B ~~ A ) -> ( f " x ) ~~ A )
33	27, 31, 32	syl2anc 643	. . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ A )
34		fin4i 8134	. . . . . . . . 9 |- ( ( ( f " x ) C. A /\ ( f " x ) ~~ A ) -> -. A e. FinIV )
35	20, 33, 34	syl2anc 643	. . . . . . . 8 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> -. A e. FinIV )
36	35	ex 424	. . . . . . 7 |- ( f : B -1-1-onto-> A -> ( ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
37	36	exlimdv 1643	. . . . . 6 |- ( f : B -1-1-onto-> A -> ( E. x ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
38	37	con2d 109	. . . . 5 |- ( f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
39	38	exlimiv 1641	. . . 4 |- ( E. f f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
40	2, 39	sylbi 188	. . 3 |- ( B ~~ A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
41		relen 7073	. . . . 5 |- Rel ~~
42	41	brrelexi 4877	. . . 4 |- ( B ~~ A -> B e. _V )
43		isfin4 8133	. . . 4 |- ( B e. _V -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
44	42, 43	syl 16	. . 3 |- ( B ~~ A -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
45	40, 44	sylibrd 226	. 2 |- ( B ~~ A -> ( A e. FinIV -> B e. FinIV ) )
46	1, 45	syl 16	1 |- ( A ~~ B -> ( A e. FinIV -> B e. FinIV ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   _Vcvv 2916   C_ wss 3280   C. wpss 3281   class class class wbr 4172   dom cdm 4837   ran crn 4838   "cima 4840   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ~~ cen 7065  Fin^IVcfin4 8116

This theorem is referenced by:  domfin4  8147  isfin4-3  8151

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 6864  df-en 7069  df-fin4 8123