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Theorem fin4en1 8474
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 7354 . 2 |- ( A ~~ B -> B ~~ A )
2 bren 7315 . . . 4 |- ( B ~~ A <-> E. f f : B -1-1-onto-> A )
3 simpr 458 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> x C. B )
4 f1of1 5637 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> f : B -1-1-> A )
5 pssss 3448 . . . . . . . . . . . . . 14 |- ( x C. B -> x C_ B )
6 ssid 3372 . . . . . . . . . . . . . 14 |- B C_ B
75, 6jctir 535 . . . . . . . . . . . . 13 |- ( x C. B -> ( x C_ B /\ B C_ B ) )
8 f1imapss 5976 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ ( x C_ B /\ B C_ B ) ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
94, 7, 8syl2an 474 . . . . . . . . . . . 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 5176 . . . . . . . . . . . . . 14 |- ( f " dom f ) = ran f
12 f1odm 5642 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A -> dom f = B )
1312imaeq2d 5166 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ( f " dom f ) = ( f " B ) )
14 dff1o5 5647 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A <-> ( f : B -1-1-> A /\ ran f = A ) )
1514simprbi 461 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ran f = A )
1611, 13, 153eqtr3a 2497 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> ( f " B ) = A )
1716adantr 462 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " B ) = A )
1817psseq2d 3446 . . . . . . . . . . 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 711 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) C. A )
21 vex 2973 . . . . . . . . . . . . . 14 |- x e. _V
2221f1imaen 7367 . . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ x C_ B ) -> ( f " x ) ~~ x )
234, 5, 22syl2an 474 . . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) ~~ x )
2423adantrr 711 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ x )
25 simprr 751 . . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> x ~~ B )
26 entr 7357 . . . . . . . . . . 11 |- ( ( ( f " x ) ~~ x /\ x ~~ B ) -> ( f " x ) ~~ B )
2724, 25, 26syl2anc 656 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ B )
28 vex 2973 . . . . . . . . . . . 12 |- f e. _V
29 f1oen3g 7321 . . . . . . . . . . . 12 |- ( ( f e. _V /\ f : B -1-1-onto-> A ) -> B ~~ A )
3028, 29mpan 665 . . . . . . . . . . 11 |- ( f : B -1-1-onto-> A -> B ~~ A )
3130adantr 462 . . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> B ~~ A )
32 entr 7357 . . . . . . . . . 10 |- ( ( ( f " x ) ~~ B /\ B ~~ A ) -> ( f " x ) ~~ A )
3327, 31, 32syl2anc 656 . . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ A )
34 fin4i 8463 . . . . . . . . 9 |- ( ( ( f " x ) C. A /\ ( f " x ) ~~ A ) -> -. A e. FinIV )
3520, 33, 34syl2anc 656 . . . . . . . 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 1695 . . . . . 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 1693 . . . 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 7311 . . . . 5 |- Rel ~~
4241brrelexi 4875 . . . 4 |- ( B ~~ A -> B e. _V )
43 isfin4 8462 . . . 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 1364   E.wex 1591   e. wcel 1761   _Vcvv 2970   C_ wss 3325   C. wpss 3326   class class class wbr 4289   dom cdm 4836   ran crn 4837   "cima 4839   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ~~ cen 7303  FinIVcfin4 8445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-er 7097  df-en 7307  df-fin4 8452
This theorem is referenced by:  domfin4  8476  isfin4-3  8480
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