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Theorem fin1a2lem7 8571
Description: Lemma for fin1a2 8580. 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 6494 . . . . . 6 |- (/) e. om
2 ne0i 3640 . . . . . 6 |- ( (/) e. om -> om =/= (/) )
3 brwdomn0 7780 . . . . . 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 5186 . . . . . . . . . 10 |- ( `' f " ran E ) C_ dom f
6 fof 5617 . . . . . . . . . . 11 |- ( f : A -onto-> om -> f : A --> om )
7 fdm 5560 . . . . . . . . . . 11 |- ( f : A --> om -> dom f = A )
86, 7syl 16 . . . . . . . . . 10 |- ( f : A -onto-> om -> dom f = A )
95, 8syl5sseq 3401 . . . . . . . . 9 |- ( f : A -onto-> om -> ( `' f " ran E ) C_ A )
10 vex 2973 . . . . . . . . . . 11 |- f e. _V
11 dmfex 6534 . . . . . . . . . . 11 |- ( ( f e. _V /\ f : A --> om ) -> A e. _V )
1210, 6, 11sylancr 658 . . . . . . . . . 10 |- ( f : A -onto-> om -> A e. _V )
13 elpw2g 4452 . . . . . . . . . 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 8568 . . . . . . . . . . . . 13 |- E : om -1-1-> om
18 f1cnv 5661 . . . . . . . . . . . . 13 |- ( E : om -1-1-> om -> `' E : ran E -1-1-onto-> om )
19 f1ofo 5645 . . . . . . . . . . . . 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 5618 . . . . . . . . . . . 12 |- ( `' E : ran E -onto-> om -> Fun `' E )
2220, 21ax-mp 5 . . . . . . . . . . 11 |- Fun `' E
2310resex 5147 . . . . . . . . . . 11 |- ( f |` ( `' f " ran E ) ) e. _V
24 cofunexg 6540 . . . . . . . . . . 11 |- ( ( Fun `' E /\ ( f |` ( `' f " ran E ) ) e. _V ) -> ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V )
2522, 23, 24mp2an 667 . . . . . . . . . 10 |- ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V
26 fofun 5618 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> Fun f )
27 fores 5626 . . . . . . . . . . . . 13 |- ( ( Fun f /\ ( `' f " ran E ) C_ dom f ) -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
2826, 5, 27sylancl 657 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
29 f1f 5603 . . . . . . . . . . . . . . 15 |- ( E : om -1-1-> om -> E : om --> om )
30 frn 5562 . . . . . . . . . . . . . . 15 |- ( E : om --> om -> ran E C_ om )
3117, 29, 30mp2b 10 . . . . . . . . . . . . . 14 |- ran E C_ om
32 foimacnv 5655 . . . . . . . . . . . . . 14 |- ( ( f : A -onto-> om /\ ran E C_ om ) -> ( f " ( `' f " ran E ) ) = ran E )
3331, 32mpan2 666 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( `' f " ran E ) ) = ran E )
34 foeq3 5615 . . . . . . . . . . . . 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 5627 . . . . . . . . . . 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 658 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( `' E o. ( f |` ( `' f " ran E ) ) ) : ( `' f " ran E ) -onto-> om )
39 fowdom 7782 . . . . . . . . . 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 658 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( `' f " ran E ) )
41 cnvexg 6523 . . . . . . . . . . . 12 |- ( f e. _V -> `' f e. _V )
42 imaexg 6514 . . . . . . . . . . . 12 |- ( `' f e. _V -> ( `' f " ran E ) e. _V )
4310, 41, 42mp2b 10 . . . . . . . . . . 11 |- ( `' f " ran E ) e. _V
44 isfin3-2 8532 . . . . . . . . . . 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 8570 . . . . . . . . . . . . . 14 |- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )
50 f1ocnv 5650 . . . . . . . . . . . . . 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 5645 . . . . . . . . . . . . . 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 5627 . . . . . . . . . . . . 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 667 . . . . . . . . . . . 12 |- ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om
55 fofun 5618 . . . . . . . . . . . 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 5147 . . . . . . . . . . 11 |- ( f |` ( A \ ( `' f " ran E ) ) ) e. _V
58 cofunexg 6540 . . . . . . . . . . 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 667 . . . . . . . . . 10 |- ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) e. _V
60 difss 3480 . . . . . . . . . . . . . 14 |- ( A \ ( `' f " ran E ) ) C_ A
6160, 8syl5sseqr 3402 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( A \ ( `' f " ran E ) ) C_ dom f )
62 fores 5626 . . . . . . . . . . . . 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 656 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( f " ( A \ ( `' f " ran E ) ) ) )
64 funcnvcnv 5473 . . . . . . . . . . . . . . . 16 |- ( Fun f -> Fun `' `' f )
65 imadif 5490 . . . . . . . . . . . . . . . 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 5166 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) )
68 difss 3480 . . . . . . . . . . . . . . 15 |- ( om \ ran E ) C_ om
69 foimacnv 5655 . . . . . . . . . . . . . . 15 |- ( ( f : A -onto-> om /\ ( om \ ran E ) C_ om ) -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
7068, 69mpan2 666 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
71 fimacnv 5832 . . . . . . . . . . . . . . . . 17 |- ( f : A --> om -> ( `' f " om ) = A )
726, 71syl 16 . . . . . . . . . . . . . . . 16 |- ( f : A -onto-> om -> ( `' f " om ) = A )
7372difeq1d 3470 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( ( `' f " om ) \ ( `' f " ran E ) ) = ( A \ ( `' f " ran E ) ) )
7473imaeq2d 5166 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) = ( f " ( A \ ( `' f " ran E ) ) ) )
7567, 70, 743eqtr3rd 2482 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( A \ ( `' f " ran E ) ) ) = ( om \ ran E ) )
76 foeq3 5615 . . . . . . . . . . . . 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 5627 . . . . . . . . . . 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 658 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> om )
81 fowdom 7782 . . . . . . . . . 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 658 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( A \ ( `' f " ran E ) ) )
83 difexg 4437 . . . . . . . . . . 11 |- ( A e. _V -> ( A \ ( `' f " ran E ) ) e. _V )
84 isfin3-2 8532 . . . . . . . . . . 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 2501 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( y e. FinIII <-> ( `' f " ran E ) e. FinIII ) )
89 difeq2 3465 . . . . . . . . . . . . 13 |- ( y = ( `' f " ran E ) -> ( A \ y ) = ( A \ ( `' f " ran E ) ) )
9089eleq1d 2507 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( ( A \ y ) e. FinIII <-> ( A \ ( `' f " ran E ) ) e. FinIII ) )
9188, 90orbi12d 704 . . . . . . . . . . 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 487 . . . . . . . . . 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 3070 . . . . . . . 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 1211 . . . . . . 7 |- ( f : A -onto-> om -> E. y e. ~P A -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
97 rexnal 2724 . . . . . . 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 1693 . . . . 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 8532 . . 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 1364   E.wex 1591   e. wcel 1761   =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970   \ cdif 3322   C_ wss 3325   (/)c0 3634   ~Pcpw 3857   class class class wbr 4289   e. cmpt 4347   Oncon0 4715   suc csuc 4717   `'ccnv 4835   dom cdm 4836   ran crn 4837   |` cres 4838   "cima 4839   o. ccom 4840   Fun wfun 5409   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414  (class class class)co 6090   omcom 6475   2oc2o 6910   .o comu 6914   ~<_* cwdom 7768  FinIIIcfin3 8446
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-3or 961  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-rmo 2721  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-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  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-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-seqom 6899  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-wdom 7770  df-card 8105  df-fin4 8452  df-fin3 8453
This theorem is referenced by:  fin1a2lem8  8572
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