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Theorem fin4en1 8685
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 7561 . 2 |- ( A ~~ B -> B ~~ A )
2 bren 7522 . . . 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 5813 . . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> f : B -1-1-> A )
5 pssss 3599 . . . . . . . . . . . . . 14 |- ( x C. B -> x C_ B )
6 ssid 3523 . . . . . . . . . . . . . 14 |- B C_ B
75, 6jctir 538 . . . . . . . . . . . . 13 |- ( x C. B -> ( x C_ B /\ B C_ B ) )
8 f1imapss 6160 . . . . . . . . . . . . 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 5345 . . . . . . . . . . . . . 14 |- ( f " dom f ) = ran f
12 f1odm 5818 . . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A -> dom f = B )
1312imaeq2d 5335 . . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ( f " dom f ) = ( f " B ) )
14 dff1o5 5823 . . . . . . . . . . . . . . 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 2532 . . . . . . . . . . . . 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 3597 . . . . . . . . . . 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 3116 . . . . . . . . . . . . . 14 |- x e. _V
2221f1imaen 7574 . . . . . . . . . . . . 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 7564 . . . . . . . . . . 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 3116 . . . . . . . . . . . 12 |- f e. _V
29 f1oen3g 7528 . . . . . . . . . . . 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 7564 . . . . . . . . . 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 8674 . . . . . . . . 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 1700 . . . . . 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 1698 . . . 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 7518 . . . . 5 |- Rel ~~
4241brrelexi 5039 . . . 4 |- ( B ~~ A -> B e. _V )
43 isfin4 8673 . . . 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 1379   E.wex 1596   e. wcel 1767   _Vcvv 3113   C_ wss 3476   C. wpss 3477   class class class wbr 4447   dom cdm 4999   ran crn 5000   "cima 5002   -1-1->wf1 5583   -1-1-onto->wf1o 5585   ~~ cen 7510  FinIVcfin4 8656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-fin4 8663
This theorem is referenced by:  domfin4  8687  isfin4-3  8691
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