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Theorem fin4en1 8757
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 7636 . 2 |- ( A ~~ B -> B ~~ A )
2 bren 7596 . . . 4 |- ( B ~~ A <-> E. f f : B -1-1-onto-> A )
3 simpr 468 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> x C. B )
4 f1of1 5827 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> f : B -1-1-> A )
5 pssss 3514 . . . . . . . . . . . . . 14 |- ( x C. B -> x C_ B )
6 ssid 3437 . . . . . . . . . . . . . 14 |- B C_ B
75, 6jctir 547 . . . . . . . . . . . . 13 |- ( x C. B -> ( x C_ B /\ B C_ B ) )
8 f1imapss 6185 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ ( x C_ B /\ B C_ B ) ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
94, 7, 8syl2an 485 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
103, 9mpbird 240 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. ( f " B ) )
11 imadmrn 5184 . . . . . . . . . . . . . 14 |- ( f " dom f ) = ran f
12 f1odm 5832 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A -> dom f = B )
1312imaeq2d 5174 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ( f " dom f ) = ( f " B ) )
14 dff1o5 5837 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A <-> ( f : B -1-1-> A /\ ran f = A ) )
1514simprbi 471 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ran f = A )
1611, 13, 153eqtr3a 2529 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> ( f " B ) = A )
1716adantr 472 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " B ) = A )
1817psseq2d 3512 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> ( f " x ) C. A ) )
1910, 18mpbid 215 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. A )
2019adantrr 731 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) C. A )
21 vex 3034 . . . . . . . . . . . . . 14 |- x e. _V
2221f1imaen 7649 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ x C_ B ) -> ( f " x ) ~~ x )
234, 5, 22syl2an 485 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) ~~ x )
2423adantrr 731 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ x )
25 simprr 774 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> x ~~ B )
26 entr 7639 . . . . . . . . . . 11 |- ( ( ( f " x ) ~~ x /\ x ~~ B ) -> ( f " x ) ~~ B )
2724, 25, 26syl2anc 673 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ B )
28 vex 3034 . . . . . . . . . . . 12 |- f e. _V
29 f1oen3g 7603 . . . . . . . . . . . 12 |- ( ( f e. _V /\ f : B -1-1-onto-> A ) -> B ~~ A )
3028, 29mpan 684 . . . . . . . . . . 11 |- ( f : B -1-1-onto-> A -> B ~~ A )
3130adantr 472 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> B ~~ A )
32 entr 7639 . . . . . . . . . 10 |- ( ( ( f " x ) ~~ B /\ B ~~ A ) -> ( f " x ) ~~ A )
3327, 31, 32syl2anc 673 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ A )
34 fin4i 8746 . . . . . . . . 9 |- ( ( ( f " x ) C. A /\ ( f " x ) ~~ A ) -> -. A e. FinIV )
3520, 33, 34syl2anc 673 . . . . . . . 8 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> -. A e. FinIV )
3635ex 441 . . . . . . 7 |- ( f : B -1-1-onto-> A -> ( ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
3736exlimdv 1787 . . . . . 6 |- ( f : B -1-1-onto-> A -> ( E. x ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
3837con2d 119 . . . . 5 |- ( f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
3938exlimiv 1784 . . . 4 |- ( E. f f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
402, 39sylbi 200 . . 3 |- ( B ~~ A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
41 relen 7592 . . . . 5 |- Rel ~~
4241brrelexi 4880 . . . 4 |- ( B ~~ A -> B e. _V )
43 isfin4 8745 . . . 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 242 . 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 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   C_ wss 3390   C. wpss 3391   class class class wbr 4395   dom cdm 4839   ran crn 4840   "cima 4842   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ~~ cen 7584  FinIVcfin4 8728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588  df-fin4 8735
This theorem is referenced by:  domfin4  8759  isfin4-3  8763
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