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Theorem fin4en1 8483
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 7363 . 2 |- ( A ~~ B -> B ~~ A )
2 bren 7324 . . . 4 |- ( B ~~ A <-> E. f f : B -1-1-onto-> A )
3 simpr 461 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> x C. B )
4 f1of1 5645 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> f : B -1-1-> A )
5 pssss 3456 . . . . . . . . . . . . . 14 |- ( x C. B -> x C_ B )
6 ssid 3380 . . . . . . . . . . . . . 14 |- B C_ B
75, 6jctir 538 . . . . . . . . . . . . 13 |- ( x C. B -> ( x C_ B /\ B C_ B ) )
8 f1imapss 5984 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ ( x C_ B /\ B C_ B ) ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
94, 7, 8syl2an 477 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
103, 9mpbird 232 . . . . . . . . . . 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 5650 . . . . . . . . . . . . . . 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 5655 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A <-> ( f : B -1-1-> A /\ ran f = A ) )
1514simprbi 464 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ran f = A )
1611, 13, 153eqtr3a 2499 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> ( f " B ) = A )
1716adantr 465 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " B ) = A )
1817psseq2d 3454 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> ( f " x ) C. A ) )
1910, 18mpbid 210 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. A )
2019adantrr 716 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) C. A )
21 vex 2980 . . . . . . . . . . . . . 14 |- x e. _V
2221f1imaen 7376 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ x C_ B ) -> ( f " x ) ~~ x )
234, 5, 22syl2an 477 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) ~~ x )
2423adantrr 716 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ x )
25 simprr 756 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> x ~~ B )
26 entr 7366 . . . . . . . . . . 11 |- ( ( ( f " x ) ~~ x /\ x ~~ B ) -> ( f " x ) ~~ B )
2724, 25, 26syl2anc 661 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ B )
28 vex 2980 . . . . . . . . . . . 12 |- f e. _V
29 f1oen3g 7330 . . . . . . . . . . . 12 |- ( ( f e. _V /\ f : B -1-1-onto-> A ) -> B ~~ A )
3028, 29mpan 670 . . . . . . . . . . 11 |- ( f : B -1-1-onto-> A -> B ~~ A )
3130adantr 465 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> B ~~ A )
32 entr 7366 . . . . . . . . . 10 |- ( ( ( f " x ) ~~ B /\ B ~~ A ) -> ( f " x ) ~~ A )
3327, 31, 32syl2anc 661 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ A )
34 fin4i 8472 . . . . . . . . 9 |- ( ( ( f " x ) C. A /\ ( f " x ) ~~ A ) -> -. A e. FinIV )
3520, 33, 34syl2anc 661 . . . . . . . 8 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> -. A e. FinIV )
3635ex 434 . . . . . . 7 |- ( f : B -1-1-onto-> A -> ( ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
3736exlimdv 1690 . . . . . 6 |- ( f : B -1-1-onto-> A -> ( E. x ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
3837con2d 115 . . . . 5 |- ( f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
3938exlimiv 1688 . . . 4 |- ( E. f f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
402, 39sylbi 195 . . 3 |- ( B ~~ A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
41 relen 7320 . . . . 5 |- Rel ~~
4241brrelexi 4884 . . . 4 |- ( B ~~ A -> B e. _V )
43 isfin4 8471 . . . 4 |- ( B e. _V -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
4442, 43syl 16 . . 3 |- ( B ~~ A -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
4540, 44sylibrd 234 . 2 |- ( B ~~ A -> ( A e. FinIV -> B e. FinIV ) )
461, 45syl 16 1 |- ( A ~~ B -> ( A e. FinIV -> B e. FinIV ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   _Vcvv 2977   C_ wss 3333   C. wpss 3334   class class class wbr 4297   dom cdm 4845   ran crn 4846   "cima 4848   -1-1->wf1 5420   -1-1-onto->wf1o 5422   ~~ cen 7312  FinIVcfin4 8454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-er 7106  df-en 7316  df-fin4 8461
This theorem is referenced by:  domfin4  8485  isfin4-3  8489
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