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Theorem fin4en1 9131
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.
StepHypRef Expression
1 ensym 8005 . 2 |- ( A ~~ B -> B ~~ A )
2 bren 7964 . . . 4 |- ( B ~~ A <-> E. f f : B -1-1-onto-> A )
3 simpr 477 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> x C. B )
4 f1of1 6136 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> f : B -1-1-> A )
5 pssss 3702 . . . . . . . . . . . . . 14 |- ( x C. B -> x C_ B )
6 ssid 3624 . . . . . . . . . . . . . 14 |- B C_ B
75, 6jctir 561 . . . . . . . . . . . . 13 |- ( x C. B -> ( x C_ B /\ B C_ B ) )
8 f1imapss 6523 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ ( x C_ B /\ B C_ B ) ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
94, 7, 8syl2an 494 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
103, 9mpbird 247 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. ( f " B ) )
11 imadmrn 5476 . . . . . . . . . . . . . 14 |- ( f " dom f ) = ran f
12 f1odm 6141 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A -> dom f = B )
1312imaeq2d 5466 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ( f " dom f ) = ( f " B ) )
14 dff1o5 6146 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A <-> ( f : B -1-1-> A /\ ran f = A ) )
1514simprbi 480 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ran f = A )
1611, 13, 153eqtr3a 2680 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> ( f " B ) = A )
1716adantr 481 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " B ) = A )
1817psseq2d 3700 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> ( f " x ) C. A ) )
1910, 18mpbid 222 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. A )
2019adantrr 753 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) C. A )
21 vex 3203 . . . . . . . . . . . . . 14 |- x e. _V
2221f1imaen 8018 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ x C_ B ) -> ( f " x ) ~~ x )
234, 5, 22syl2an 494 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) ~~ x )
2423adantrr 753 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ x )
25 simprr 796 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> x ~~ B )
26 entr 8008 . . . . . . . . . . 11 |- ( ( ( f " x ) ~~ x /\ x ~~ B ) -> ( f " x ) ~~ B )
2724, 25, 26syl2anc 693 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ B )
28 vex 3203 . . . . . . . . . . . 12 |- f e. _V
29 f1oen3g 7971 . . . . . . . . . . . 12 |- ( ( f e. _V /\ f : B -1-1-onto-> A ) -> B ~~ A )
3028, 29mpan 706 . . . . . . . . . . 11 |- ( f : B -1-1-onto-> A -> B ~~ A )
3130adantr 481 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> B ~~ A )
32 entr 8008 . . . . . . . . . 10 |- ( ( ( f " x ) ~~ B /\ B ~~ A ) -> ( f " x ) ~~ A )
3327, 31, 32syl2anc 693 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ A )
34 fin4i 9120 . . . . . . . . 9 |- ( ( ( f " x ) C. A /\ ( f " x ) ~~ A ) -> -. A e. FinIV )
3520, 33, 34syl2anc 693 . . . . . . . 8 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> -. A e. FinIV )
3635ex 450 . . . . . . 7 |- ( f : B -1-1-onto-> A -> ( ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
3736exlimdv 1861 . . . . . 6 |- ( f : B -1-1-onto-> A -> ( E. x ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
3837con2d 129 . . . . 5 |- ( f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
3938exlimiv 1858 . . . 4 |- ( E. f f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
402, 39sylbi 207 . . 3 |- ( B ~~ A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
41 relen 7960 . . . . 5 |- Rel ~~
4241brrelexi 5158 . . . 4 |- ( B ~~ A -> B e. _V )
43 isfin4 9119 . . . 4 |- ( B e. _V -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
4442, 43syl 17 . . 3 |- ( B ~~ A -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
4540, 44sylibrd 249 . 2 |- ( B ~~ A -> ( A e. FinIV -> B e. FinIV ) )
461, 45syl 17 1 |- ( A ~~ B -> ( A e. FinIV -> B e. FinIV ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   C. wpss 3575   class class class wbr 4653   dom cdm 5114   ran crn 5115   "cima 5117   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ~~ cen 7952  FinIVcfin4 9102
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-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-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-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-fin4 9109
This theorem is referenced by:  domfin4  9133  isfin4-3  9137
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