MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1a2lem7 Unicode version
Theorem fin1a2lem7 8242
Description: Lemma for fin1a2 8251. 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 4823 . . . . . 6 |- (/) e. om
2 ne0i 3594 . . . . . 6 |- ( (/) e. om -> om =/= (/) )
3 brwdomn0 7493 . . . . . 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 5183 . . . . . . . . . 10 |- ( `' f " ran E ) C_ dom f
6 fof 5612 . . . . . . . . . . 11 |- ( f : A -onto-> om -> f : A --> om )
7 fdm 5554 . . . . . . . . . . 11 |- ( f : A --> om -> dom f = A )
86, 7syl 16 . . . . . . . . . 10 |- ( f : A -onto-> om -> dom f = A )
95, 8syl5sseq 3356 . . . . . . . . 9 |- ( f : A -onto-> om -> ( `' f " ran E ) C_ A )
10 vex 2919 . . . . . . . . . . 11 |- f e. _V
11 dmfex 5585 . . . . . . . . . . 11 |- ( ( f e. _V /\ f : A --> om ) -> A e. _V )
1210, 6, 11sylancr 645 . . . . . . . . . 10 |- ( f : A -onto-> om -> A e. _V )
13 elpw2g 4323 . . . . . . . . . 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 224 . . . . . . . 8 |- ( f : A -onto-> om -> ( `' f " ran E ) e. ~P A )
16 fin1a2lem.b . . . . . . . . . . . . . 14 |- E = ( x e. om |-> ( 2o .o x ) )
1716fin1a2lem4 8239 . . . . . . . . . . . . 13 |- E : om -1-1-> om
18 f1cnv 5658 . . . . . . . . . . . . 13 |- ( E : om -1-1-> om -> `' E : ran E -1-1-onto-> om )
19 f1ofo 5640 . . . . . . . . . . . . 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 5613 . . . . . . . . . . . 12 |- ( `' E : ran E -onto-> om -> Fun `' E )
2220, 21ax-mp 8 . . . . . . . . . . 11 |- Fun `' E
2310resex 5145 . . . . . . . . . . 11 |- ( f |` ( `' f " ran E ) ) e. _V
24 cofunexg 5918 . . . . . . . . . . 11 |- ( ( Fun `' E /\ ( f |` ( `' f " ran E ) ) e. _V ) -> ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V )
2522, 23, 24mp2an 654 . . . . . . . . . 10 |- ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V
26 fofun 5613 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> Fun f )
27 fores 5621 . . . . . . . . . . . . 13 |- ( ( Fun f /\ ( `' f " ran E ) C_ dom f ) -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
2826, 5, 27sylancl 644 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
29 f1f 5598 . . . . . . . . . . . . . . 15 |- ( E : om -1-1-> om -> E : om --> om )
30 frn 5556 . . . . . . . . . . . . . . 15 |- ( E : om --> om -> ran E C_ om )
3117, 29, 30mp2b 10 . . . . . . . . . . . . . 14 |- ran E C_ om
32 foimacnv 5651 . . . . . . . . . . . . . 14 |- ( ( f : A -onto-> om /\ ran E C_ om ) -> ( f " ( `' f " ran E ) ) = ran E )
3331, 32mpan2 653 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( `' f " ran E ) ) = ran E )
34 foeq3 5610 . . . . . . . . . . . . 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 202 . . . . . . . . . . 11 |- ( f : A -onto-> om -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ran E )
37 foco 5622 . . . . . . . . . . 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 645 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( `' E o. ( f |` ( `' f " ran E ) ) ) : ( `' f " ran E ) -onto-> om )
39 fowdom 7495 . . . . . . . . . 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 645 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( `' f " ran E ) )
41 cnvexg 5364 . . . . . . . . . . . 12 |- ( f e. _V -> `' f e. _V )
42 imaexg 5176 . . . . . . . . . . . 12 |- ( `' f e. _V -> ( `' f " ran E ) e. _V )
4310, 41, 42mp2b 10 . . . . . . . . . . 11 |- ( `' f " ran E ) e. _V
44 isfin3-2 8203 . . . . . . . . . . 11 |- ( ( `' f " ran E ) e. _V -> ( ( `' f " ran E ) e. FinIII <-> -. om ~<_* ( `' f " ran E ) ) )
4543, 44ax-mp 8 . . . . . . . . . 10 |- ( ( `' f " ran E ) e. FinIII <-> -. om ~<_* ( `' f " ran E ) )
4645con2bii 323 . . . . . . . . 9 |- ( om ~<_* ( `' f " ran E ) <-> -. ( `' f " ran E ) e. FinIII )
4740, 46sylib 189 . . . . . . . 8 |- ( f : A -onto-> om -> -. ( `' f " ran E ) e. FinIII )
48 fin1a2lem.aa . . . . . . . . . . . . . . 15 |- S = ( x e. On |-> suc x )
4916, 48fin1a2lem6 8241 . . . . . . . . . . . . . 14 |- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )
50 f1ocnv 5646 . . . . . . . . . . . . . 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 5640 . . . . . . . . . . . . . 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 5622 . . . . . . . . . . . . 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 654 . . . . . . . . . . . 12 |- ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om
55 fofun 5613 . . . . . . . . . . . 12 |- ( ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om -> Fun ( `' E o. `' ( S |` ran E ) ) )
5654, 55ax-mp 8 . . . . . . . . . . 11 |- Fun ( `' E o. `' ( S |` ran E ) )
5710resex 5145 . . . . . . . . . . 11 |- ( f |` ( A \ ( `' f " ran E ) ) ) e. _V
58 cofunexg 5918 . . . . . . . . . . 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 654 . . . . . . . . . 10 |- ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) e. _V
60 difss 3434 . . . . . . . . . . . . . 14 |- ( A \ ( `' f " ran E ) ) C_ A
6160, 8syl5sseqr 3357 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( A \ ( `' f " ran E ) ) C_ dom f )
62 fores 5621 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( f " ( A \ ( `' f " ran E ) ) ) )
64 funcnvcnv 5468 . . . . . . . . . . . . . . . 16 |- ( Fun f -> Fun `' `' f )
65 imadif 5487 . . . . . . . . . . . . . . . 16 |- ( Fun `' `' f -> ( `' f " ( om \ ran E ) ) = ( ( `' f " om ) \ ( `' f " ran E ) ) )
6626, 64, 653syl 19 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( `' f " ( om \ ran E ) ) = ( ( `' f " om ) \ ( `' f " ran E ) ) )
6766imaeq2d 5162 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) )
68 difss 3434 . . . . . . . . . . . . . . 15 |- ( om \ ran E ) C_ om
69 foimacnv 5651 . . . . . . . . . . . . . . 15 |- ( ( f : A -onto-> om /\ ( om \ ran E ) C_ om ) -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
7068, 69mpan2 653 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
71 fimacnv 5821 . . . . . . . . . . . . . . . . 17 |- ( f : A --> om -> ( `' f " om ) = A )
726, 71syl 16 . . . . . . . . . . . . . . . 16 |- ( f : A -onto-> om -> ( `' f " om ) = A )
7372difeq1d 3424 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( ( `' f " om ) \ ( `' f " ran E ) ) = ( A \ ( `' f " ran E ) ) )
7473imaeq2d 5162 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) = ( f " ( A \ ( `' f " ran E ) ) ) )
7567, 70, 743eqtr3rd 2445 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( A \ ( `' f " ran E ) ) ) = ( om \ ran E ) )
76 foeq3 5610 . . . . . . . . . . . . 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 202 . . . . . . . . . . 11 |- ( f : A -onto-> om -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( om \ ran E ) )
79 foco 5622 . . . . . . . . . . 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 645 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> om )
81 fowdom 7495 . . . . . . . . . 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 645 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( A \ ( `' f " ran E ) ) )
83 difexg 4311 . . . . . . . . . . 11 |- ( A e. _V -> ( A \ ( `' f " ran E ) ) e. _V )
84 isfin3-2 8203 . . . . . . . . . . 11 |- ( ( A \ ( `' f " ran E ) ) e. _V -> ( ( A \ ( `' f " ran E ) ) e. FinIII <-> -. om ~<_* ( A \ ( `' f " ran E ) ) ) )
8512, 83, 843syl 19 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( ( A \ ( `' f " ran E ) ) e. FinIII <-> -. om ~<_* ( A \ ( `' f " ran E ) ) ) )
8685con2bid 320 . . . . . . . . 9 |- ( f : A -onto-> om -> ( om ~<_* ( A \ ( `' f " ran E ) ) <-> -. ( A \ ( `' f " ran E ) ) e. FinIII ) )
8782, 86mpbid 202 . . . . . . . 8 |- ( f : A -onto-> om -> -. ( A \ ( `' f " ran E ) ) e. FinIII )
88 eleq1 2464 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( y e. FinIII <-> ( `' f " ran E ) e. FinIII ) )
89 difeq2 3419 . . . . . . . . . . . . 13 |- ( y = ( `' f " ran E ) -> ( A \ y ) = ( A \ ( `' f " ran E ) ) )
9089eleq1d 2470 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( ( A \ y ) e. FinIII <-> ( A \ ( `' f " ran E ) ) e. FinIII ) )
9188, 90orbi12d 691 . . . . . . . . . . 11 |- ( y = ( `' f " ran E ) -> ( ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> ( ( `' f " ran E ) e. FinIII \/ ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
9291notbid 286 . . . . . . . . . 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 477 . . . . . . . . . 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 253 . . . . . . . . 9 |- ( y = ( `' f " ran E ) -> ( -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> ( -. ( `' f " ran E ) e. FinIII /\ -. ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
9594rspcev 3012 . . . . . . . 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 1182 . . . . . . 7 |- ( f : A -onto-> om -> E. y e. ~P A -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
97 rexnal 2677 . . . . . . 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 189 . . . . . 6 |- ( f : A -onto-> om -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
9998exlimiv 1641 . . . . 5 |- ( E. f f : A -onto-> om -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
1004, 99sylbi 188 . . . 4 |- ( om ~<_* A -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
101100con2i 114 . . 3 |- ( A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) -> -. om ~<_* A )
102 isfin3-2 8203 . . 3 |- ( A e. V -> ( A e. FinIII <-> -. om ~<_* A ) )
103101, 102syl5ibr 213 . 2 |- ( A e. V -> ( A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) -> A e. FinIII ) )
104103imp 419 1 |- ( ( A e. V /\ A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) ) -> A e. FinIII )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916   \ cdif 3277   C_ wss 3280   (/)c0 3588   ~Pcpw 3759   class class class wbr 4172   e. cmpt 4226   Oncon0 4541   suc csuc 4543   omcom 4804   `'ccnv 4836   dom cdm 4837   ran crn 4838   |` cres 4839   "cima 4840   o. ccom 4841   Fun wfun 5407   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412  (class class class)co 6040   2oc2o 6677   .o comu 6681   ~<_* cwdom 7481  FinIIIcfin3 8117
This theorem is referenced by:  fin1a2lem8  8243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-seqom 6664  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-wdom 7483  df-card 7782  df-fin4 8123  df-fin3 8124
  Copyright terms: Public domain W3C validator