Theorem fin4en1 8736

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.

Step	Hyp	Ref	Expression
1		ensym 7615	. 2 |- ( A ~~ B -> B ~~ A )
2		bren 7575	. . . 4 |- ( B ~~ A <-> E. f f : B -1-1-onto-> A )
3		simpr 463	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> x C. B )
4		f1of1 5811	. . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> f : B -1-1-> A )
5		pssss 3527	. . . . . . . . . . . . . 14 |- ( x C. B -> x C_ B )
6		ssid 3450	. . . . . . . . . . . . . 14 |- B C_ B
7	5, 6	jctir 541	. . . . . . . . . . . . 13 |- ( x C. B -> ( x C_ B /\ B C_ B ) )
8		f1imapss 6165	. . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ ( x C_ B /\ B C_ B ) ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
9	4, 7, 8	syl2an 480	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> x C. B ) )
10	3, 9	mpbird 236	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. ( f " B ) )
11		imadmrn 5177	. . . . . . . . . . . . . 14 |- ( f " dom f ) = ran f
12		f1odm 5816	. . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A -> dom f = B )
13	12	imaeq2d 5167	. . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ( f " dom f ) = ( f " B ) )
14		dff1o5 5821	. . . . . . . . . . . . . . 15 |- ( f : B -1-1-onto-> A <-> ( f : B -1-1-> A /\ ran f = A ) )
15	14	simprbi 466	. . . . . . . . . . . . . 14 |- ( f : B -1-1-onto-> A -> ran f = A )
16	11, 13, 15	3eqtr3a 2508	. . . . . . . . . . . . 13 |- ( f : B -1-1-onto-> A -> ( f " B ) = A )
17	16	adantr 467	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " B ) = A )
18	17	psseq2d 3525	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( ( f " x ) C. ( f " B ) <-> ( f " x ) C. A ) )
19	10, 18	mpbid 214	. . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) C. A )
20	19	adantrr 722	. . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) C. A )
21		vex 3047	. . . . . . . . . . . . . 14 |- x e. _V
22	21	f1imaen 7628	. . . . . . . . . . . . 13 |- ( ( f : B -1-1-> A /\ x C_ B ) -> ( f " x ) ~~ x )
23	4, 5, 22	syl2an 480	. . . . . . . . . . . 12 |- ( ( f : B -1-1-onto-> A /\ x C. B ) -> ( f " x ) ~~ x )
24	23	adantrr 722	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ x )
25		simprr 765	. . . . . . . . . . 11 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> x ~~ B )
26		entr 7618	. . . . . . . . . . 11 |- ( ( ( f " x ) ~~ x /\ x ~~ B ) -> ( f " x ) ~~ B )
27	24, 25, 26	syl2anc 666	. . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ B )
28		vex 3047	. . . . . . . . . . . 12 |- f e. _V
29		f1oen3g 7582	. . . . . . . . . . . 12 |- ( ( f e. _V /\ f : B -1-1-onto-> A ) -> B ~~ A )
30	28, 29	mpan 675	. . . . . . . . . . 11 |- ( f : B -1-1-onto-> A -> B ~~ A )
31	30	adantr 467	. . . . . . . . . 10 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> B ~~ A )
32		entr 7618	. . . . . . . . . 10 |- ( ( ( f " x ) ~~ B /\ B ~~ A ) -> ( f " x ) ~~ A )
33	27, 31, 32	syl2anc 666	. . . . . . . . 9 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> ( f " x ) ~~ A )
34		fin4i 8725	. . . . . . . . 9 |- ( ( ( f " x ) C. A /\ ( f " x ) ~~ A ) -> -. A e. FinIV )
35	20, 33, 34	syl2anc 666	. . . . . . . 8 |- ( ( f : B -1-1-onto-> A /\ ( x C. B /\ x ~~ B ) ) -> -. A e. FinIV )
36	35	ex 436	. . . . . . 7 |- ( f : B -1-1-onto-> A -> ( ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
37	36	exlimdv 1778	. . . . . 6 |- ( f : B -1-1-onto-> A -> ( E. x ( x C. B /\ x ~~ B ) -> -. A e. FinIV ) )
38	37	con2d 119	. . . . 5 |- ( f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
39	38	exlimiv 1775	. . . 4 |- ( E. f f : B -1-1-onto-> A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
40	2, 39	sylbi 199	. . 3 |- ( B ~~ A -> ( A e. FinIV -> -. E. x ( x C. B /\ x ~~ B ) ) )
41		relen 7571	. . . . 5 |- Rel ~~
42	41	brrelexi 4874	. . . 4 |- ( B ~~ A -> B e. _V )
43		isfin4 8724	. . . 4 |- ( B e. _V -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
44	42, 43	syl 17	. . 3 |- ( B ~~ A -> ( B e. FinIV <-> -. E. x ( x C. B /\ x ~~ B ) ) )
45	40, 44	sylibrd 238	. 2 |- ( B ~~ A -> ( A e. FinIV -> B e. FinIV ) )
46	1, 45	syl 17	1 |- ( A ~~ B -> ( A e. FinIV -> B e. FinIV ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   E.wex 1662   e. wcel 1886   _Vcvv 3044   C_ wss 3403   C. wpss 3404   class class class wbr 4401   dom cdm 4833   ran crn 4834   "cima 4836   -1-1->wf1 5578   -1-1-onto->wf1o 5580   ~~ cen 7563  Fin^IVcfin4 8707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-er 7360  df-en 7567  df-fin4 8714

This theorem is referenced by:  domfin4  8738  isfin4-3  8742