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Theorem fin1a2lem7 8782
Description: Lemma for fin1a2 8791. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b |- E = ( x e. om |-> ( 2o .o x ) )
fin1a2lem.aa |- S = ( x e. On |-> suc x )
Assertion
Ref Expression
fin1a2lem7 |- ( ( A e. V /\ A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) ) -> A e. FinIII )
Distinct variable groups:   y, A   y, E
Allowed substitution hints:   A( x)   S( x, y)   E( x)   V( x, y)
Proof of Theorem fin1a2lem7
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 peano1 6697 . . . . . 6 |- (/) e. om
2 ne0i 3791 . . . . . 6 |- ( (/) e. om -> om =/= (/) )
3 brwdomn0 7991 . . . . . 6 |- ( om =/= (/) -> ( om ~<_* A <-> E. f f : A -onto-> om ) )
41, 2, 3mp2b 10 . . . . 5 |- ( om ~<_* A <-> E. f f : A -onto-> om )
5 cnvimass 5355 . . . . . . . . . 10 |- ( `' f " ran E ) C_ dom f
6 fof 5793 . . . . . . . . . . 11 |- ( f : A -onto-> om -> f : A --> om )
7 fdm 5733 . . . . . . . . . . 11 |- ( f : A --> om -> dom f = A )
86, 7syl 16 . . . . . . . . . 10 |- ( f : A -onto-> om -> dom f = A )
95, 8syl5sseq 3552 . . . . . . . . 9 |- ( f : A -onto-> om -> ( `' f " ran E ) C_ A )
10 vex 3116 . . . . . . . . . . 11 |- f e. _V
11 dmfex 6739 . . . . . . . . . . 11 |- ( ( f e. _V /\ f : A --> om ) -> A e. _V )
1210, 6, 11sylancr 663 . . . . . . . . . 10 |- ( f : A -onto-> om -> A e. _V )
13 elpw2g 4610 . . . . . . . . . 10 |- ( A e. _V -> ( ( `' f " ran E ) e. ~P A <-> ( `' f " ran E ) C_ A ) )
1412, 13syl 16 . . . . . . . . 9 |- ( f : A -onto-> om -> ( ( `' f " ran E ) e. ~P A <-> ( `' f " ran E ) C_ A ) )
159, 14mpbird 232 . . . . . . . 8 |- ( f : A -onto-> om -> ( `' f " ran E ) e. ~P A )
16 fin1a2lem.b . . . . . . . . . . . . . 14 |- E = ( x e. om |-> ( 2o .o x ) )
1716fin1a2lem4 8779 . . . . . . . . . . . . 13 |- E : om -1-1-> om
18 f1cnv 5837 . . . . . . . . . . . . 13 |- ( E : om -1-1-> om -> `' E : ran E -1-1-onto-> om )
19 f1ofo 5821 . . . . . . . . . . . . 13 |- ( `' E : ran E -1-1-onto-> om -> `' E : ran E -onto-> om )
2017, 18, 19mp2b 10 . . . . . . . . . . . 12 |- `' E : ran E -onto-> om
21 fofun 5794 . . . . . . . . . . . 12 |- ( `' E : ran E -onto-> om -> Fun `' E )
2220, 21ax-mp 5 . . . . . . . . . . 11 |- Fun `' E
2310resex 5315 . . . . . . . . . . 11 |- ( f |` ( `' f " ran E ) ) e. _V
24 cofunexg 6745 . . . . . . . . . . 11 |- ( ( Fun `' E /\ ( f |` ( `' f " ran E ) ) e. _V ) -> ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V )
2522, 23, 24mp2an 672 . . . . . . . . . 10 |- ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V
26 fofun 5794 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> Fun f )
27 fores 5802 . . . . . . . . . . . . 13 |- ( ( Fun f /\ ( `' f " ran E ) C_ dom f ) -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
2826, 5, 27sylancl 662 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
29 f1f 5779 . . . . . . . . . . . . . . 15 |- ( E : om -1-1-> om -> E : om --> om )
30 frn 5735 . . . . . . . . . . . . . . 15 |- ( E : om --> om -> ran E C_ om )
3117, 29, 30mp2b 10 . . . . . . . . . . . . . 14 |- ran E C_ om
32 foimacnv 5831 . . . . . . . . . . . . . 14 |- ( ( f : A -onto-> om /\ ran E C_ om ) -> ( f " ( `' f " ran E ) ) = ran E )
3331, 32mpan2 671 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( `' f " ran E ) ) = ran E )
34 foeq3 5791 . . . . . . . . . . . . 13 |- ( ( f " ( `' f " ran E ) ) = ran E -> ( ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) <-> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ran E ) )
3533, 34syl 16 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) <-> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ran E ) )
3628, 35mpbid 210 . . . . . . . . . . 11 |- ( f : A -onto-> om -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ran E )
37 foco 5803 . . . . . . . . . . 11 |- ( ( `' E : ran E -onto-> om /\ ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ran E ) -> ( `' E o. ( f |` ( `' f " ran E ) ) ) : ( `' f " ran E ) -onto-> om )
3820, 36, 37sylancr 663 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( `' E o. ( f |` ( `' f " ran E ) ) ) : ( `' f " ran E ) -onto-> om )
39 fowdom 7993 . . . . . . . . . 10 |- ( ( ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V /\ ( `' E o. ( f |` ( `' f " ran E ) ) ) : ( `' f " ran E ) -onto-> om ) -> om ~<_* ( `' f " ran E ) )
4025, 38, 39sylancr 663 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( `' f " ran E ) )
41 cnvexg 6727 . . . . . . . . . . . 12 |- ( f e. _V -> `' f e. _V )
42 imaexg 6718 . . . . . . . . . . . 12 |- ( `' f e. _V -> ( `' f " ran E ) e. _V )
4310, 41, 42mp2b 10 . . . . . . . . . . 11 |- ( `' f " ran E ) e. _V
44 isfin3-2 8743 . . . . . . . . . . 11 |- ( ( `' f " ran E ) e. _V -> ( ( `' f " ran E ) e. FinIII <-> -. om ~<_* ( `' f " ran E ) ) )
4543, 44ax-mp 5 . . . . . . . . . 10 |- ( ( `' f " ran E ) e. FinIII <-> -. om ~<_* ( `' f " ran E ) )
4645con2bii 332 . . . . . . . . 9 |- ( om ~<_* ( `' f " ran E ) <-> -. ( `' f " ran E ) e. FinIII )
4740, 46sylib 196 . . . . . . . 8 |- ( f : A -onto-> om -> -. ( `' f " ran E ) e. FinIII )
48 fin1a2lem.aa . . . . . . . . . . . . . . 15 |- S = ( x e. On |-> suc x )
4916, 48fin1a2lem6 8781 . . . . . . . . . . . . . 14 |- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )
50 f1ocnv 5826 . . . . . . . . . . . . . 14 |- ( ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E ) -> `' ( S |` ran E ) : ( om \ ran E ) -1-1-onto-> ran E )
51 f1ofo 5821 . . . . . . . . . . . . . 14 |- ( `' ( S |` ran E ) : ( om \ ran E ) -1-1-onto-> ran E -> `' ( S |` ran E ) : ( om \ ran E ) -onto-> ran E )
5249, 50, 51mp2b 10 . . . . . . . . . . . . 13 |- `' ( S |` ran E ) : ( om \ ran E ) -onto-> ran E
53 foco 5803 . . . . . . . . . . . . 13 |- ( ( `' E : ran E -onto-> om /\ `' ( S |` ran E ) : ( om \ ran E ) -onto-> ran E ) -> ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om )
5420, 52, 53mp2an 672 . . . . . . . . . . . 12 |- ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om
55 fofun 5794 . . . . . . . . . . . 12 |- ( ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om -> Fun ( `' E o. `' ( S |` ran E ) ) )
5654, 55ax-mp 5 . . . . . . . . . . 11 |- Fun ( `' E o. `' ( S |` ran E ) )
5710resex 5315 . . . . . . . . . . 11 |- ( f |` ( A \ ( `' f " ran E ) ) ) e. _V
58 cofunexg 6745 . . . . . . . . . . 11 |- ( ( Fun ( `' E o. `' ( S |` ran E ) ) /\ ( f |` ( A \ ( `' f " ran E ) ) ) e. _V ) -> ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) e. _V )
5956, 57, 58mp2an 672 . . . . . . . . . 10 |- ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) e. _V
60 difss 3631 . . . . . . . . . . . . . 14 |- ( A \ ( `' f " ran E ) ) C_ A
6160, 8syl5sseqr 3553 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( A \ ( `' f " ran E ) ) C_ dom f )
62 fores 5802 . . . . . . . . . . . . 13 |- ( ( Fun f /\ ( A \ ( `' f " ran E ) ) C_ dom f ) -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( f " ( A \ ( `' f " ran E ) ) ) )
6326, 61, 62syl2anc 661 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( f " ( A \ ( `' f " ran E ) ) ) )
64 funcnvcnv 5644 . . . . . . . . . . . . . . . 16 |- ( Fun f -> Fun `' `' f )
65 imadif 5661 . . . . . . . . . . . . . . . 16 |- ( Fun `' `' f -> ( `' f " ( om \ ran E ) ) = ( ( `' f " om ) \ ( `' f " ran E ) ) )
6626, 64, 653syl 20 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( `' f " ( om \ ran E ) ) = ( ( `' f " om ) \ ( `' f " ran E ) ) )
6766imaeq2d 5335 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) )
68 difss 3631 . . . . . . . . . . . . . . 15 |- ( om \ ran E ) C_ om
69 foimacnv 5831 . . . . . . . . . . . . . . 15 |- ( ( f : A -onto-> om /\ ( om \ ran E ) C_ om ) -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
7068, 69mpan2 671 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
71 fimacnv 6011 . . . . . . . . . . . . . . . . 17 |- ( f : A --> om -> ( `' f " om ) = A )
726, 71syl 16 . . . . . . . . . . . . . . . 16 |- ( f : A -onto-> om -> ( `' f " om ) = A )
7372difeq1d 3621 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( ( `' f " om ) \ ( `' f " ran E ) ) = ( A \ ( `' f " ran E ) ) )
7473imaeq2d 5335 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) = ( f " ( A \ ( `' f " ran E ) ) ) )
7567, 70, 743eqtr3rd 2517 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( A \ ( `' f " ran E ) ) ) = ( om \ ran E ) )
76 foeq3 5791 . . . . . . . . . . . . 13 |- ( ( f " ( A \ ( `' f " ran E ) ) ) = ( om \ ran E ) -> ( ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( f " ( A \ ( `' f " ran E ) ) ) <-> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( om \ ran E ) ) )
7775, 76syl 16 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( f " ( A \ ( `' f " ran E ) ) ) <-> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( om \ ran E ) ) )
7863, 77mpbid 210 . . . . . . . . . . 11 |- ( f : A -onto-> om -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( om \ ran E ) )
79 foco 5803 . . . . . . . . . . 11 |- ( ( ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om /\ ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( om \ ran E ) ) -> ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> om )
8054, 78, 79sylancr 663 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> om )
81 fowdom 7993 . . . . . . . . . 10 |- ( ( ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) e. _V /\ ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> om ) -> om ~<_* ( A \ ( `' f " ran E ) ) )
8259, 80, 81sylancr 663 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( A \ ( `' f " ran E ) ) )
83 difexg 4595 . . . . . . . . . . 11 |- ( A e. _V -> ( A \ ( `' f " ran E ) ) e. _V )
84 isfin3-2 8743 . . . . . . . . . . 11 |- ( ( A \ ( `' f " ran E ) ) e. _V -> ( ( A \ ( `' f " ran E ) ) e. FinIII <-> -. om ~<_* ( A \ ( `' f " ran E ) ) ) )
8512, 83, 843syl 20 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( ( A \ ( `' f " ran E ) ) e. FinIII <-> -. om ~<_* ( A \ ( `' f " ran E ) ) ) )
8685con2bid 329 . . . . . . . . 9 |- ( f : A -onto-> om -> ( om ~<_* ( A \ ( `' f " ran E ) ) <-> -. ( A \ ( `' f " ran E ) ) e. FinIII ) )
8782, 86mpbid 210 . . . . . . . 8 |- ( f : A -onto-> om -> -. ( A \ ( `' f " ran E ) ) e. FinIII )
88 eleq1 2539 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( y e. FinIII <-> ( `' f " ran E ) e. FinIII ) )
89 difeq2 3616 . . . . . . . . . . . . 13 |- ( y = ( `' f " ran E ) -> ( A \ y ) = ( A \ ( `' f " ran E ) ) )
9089eleq1d 2536 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( ( A \ y ) e. FinIII <-> ( A \ ( `' f " ran E ) ) e. FinIII ) )
9188, 90orbi12d 709 . . . . . . . . . . 11 |- ( y = ( `' f " ran E ) -> ( ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> ( ( `' f " ran E ) e. FinIII \/ ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
9291notbid 294 . . . . . . . . . 10 |- ( y = ( `' f " ran E ) -> ( -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> -. ( ( `' f " ran E ) e. FinIII \/ ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
93 ioran 490 . . . . . . . . . 10 |- ( -. ( ( `' f " ran E ) e. FinIII \/ ( A \ ( `' f " ran E ) ) e. FinIII ) <-> ( -. ( `' f " ran E ) e. FinIII /\ -. ( A \ ( `' f " ran E ) ) e. FinIII ) )
9492, 93syl6bb 261 . . . . . . . . 9 |- ( y = ( `' f " ran E ) -> ( -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> ( -. ( `' f " ran E ) e. FinIII /\ -. ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
9594rspcev 3214 . . . . . . . 8 |- ( ( ( `' f " ran E ) e. ~P A /\ ( -. ( `' f " ran E ) e. FinIII /\ -. ( A \ ( `' f " ran E ) ) e. FinIII ) ) -> E. y e. ~P A -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
9615, 47, 87, 95syl12anc 1226 . . . . . . 7 |- ( f : A -onto-> om -> E. y e. ~P A -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
97 rexnal 2912 . . . . . . 7 |- ( E. y e. ~P A -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
9896, 97sylib 196 . . . . . 6 |- ( f : A -onto-> om -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
9998exlimiv 1698 . . . . 5 |- ( E. f f : A -onto-> om -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
1004, 99sylbi 195 . . . 4 |- ( om ~<_* A -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
101100con2i 120 . . 3 |- ( A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) -> -. om ~<_* A )
102 isfin3-2 8743 . . 3 |- ( A e. V -> ( A e. FinIII <-> -. om ~<_* A ) )
103101, 102syl5ibr 221 . 2 |- ( A e. V -> ( A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) -> A e. FinIII ) )
104103imp 429 1 |- ( ( A e. V /\ A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) ) -> A e. FinIII )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   \ cdif 3473   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447   |-> cmpt 4505   Oncon0 4878   suc csuc 4880   `'ccnv 4998   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   o. ccom 5003   Fun wfun 5580   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585  (class class class)co 6282   omcom 6678   2oc2o 7121   .o comu 7125   ~<_* cwdom 7979  FinIIIcfin3 8657
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-3or 974  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-rmo 2822  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-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-seqom 7110  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-wdom 7981  df-card 8316  df-fin4 8663  df-fin3 8664
This theorem is referenced by:  fin1a2lem8  8783
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