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Theorem fin1a2lem7 8676
Description: Lemma for fin1a2 8685. 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 6595 . . . . . 6 |- (/) e. om
2 ne0i 3741 . . . . . 6 |- ( (/) e. om -> om =/= (/) )
3 brwdomn0 7885 . . . . . 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 5287 . . . . . . . . . 10 |- ( `' f " ran E ) C_ dom f
6 fof 5718 . . . . . . . . . . 11 |- ( f : A -onto-> om -> f : A --> om )
7 fdm 5661 . . . . . . . . . . 11 |- ( f : A --> om -> dom f = A )
86, 7syl 16 . . . . . . . . . 10 |- ( f : A -onto-> om -> dom f = A )
95, 8syl5sseq 3502 . . . . . . . . 9 |- ( f : A -onto-> om -> ( `' f " ran E ) C_ A )
10 vex 3071 . . . . . . . . . . 11 |- f e. _V
11 dmfex 6635 . . . . . . . . . . 11 |- ( ( f e. _V /\ f : A --> om ) -> A e. _V )
1210, 6, 11sylancr 663 . . . . . . . . . 10 |- ( f : A -onto-> om -> A e. _V )
13 elpw2g 4553 . . . . . . . . . 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 8673 . . . . . . . . . . . . 13 |- E : om -1-1-> om
18 f1cnv 5762 . . . . . . . . . . . . 13 |- ( E : om -1-1-> om -> `' E : ran E -1-1-onto-> om )
19 f1ofo 5746 . . . . . . . . . . . . 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 5719 . . . . . . . . . . . 12 |- ( `' E : ran E -onto-> om -> Fun `' E )
2220, 21ax-mp 5 . . . . . . . . . . 11 |- Fun `' E
2310resex 5248 . . . . . . . . . . 11 |- ( f |` ( `' f " ran E ) ) e. _V
24 cofunexg 6641 . . . . . . . . . . 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 5719 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> Fun f )
27 fores 5727 . . . . . . . . . . . . 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 5704 . . . . . . . . . . . . . . 15 |- ( E : om -1-1-> om -> E : om --> om )
30 frn 5663 . . . . . . . . . . . . . . 15 |- ( E : om --> om -> ran E C_ om )
3117, 29, 30mp2b 10 . . . . . . . . . . . . . 14 |- ran E C_ om
32 foimacnv 5756 . . . . . . . . . . . . . 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 5716 . . . . . . . . . . . . 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 5728 . . . . . . . . . . 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 7887 . . . . . . . . . 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 6624 . . . . . . . . . . . 12 |- ( f e. _V -> `' f e. _V )
42 imaexg 6615 . . . . . . . . . . . 12 |- ( `' f e. _V -> ( `' f " ran E ) e. _V )
4310, 41, 42mp2b 10 . . . . . . . . . . 11 |- ( `' f " ran E ) e. _V
44 isfin3-2 8637 . . . . . . . . . . 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 8675 . . . . . . . . . . . . . 14 |- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )
50 f1ocnv 5751 . . . . . . . . . . . . . 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 5746 . . . . . . . . . . . . . 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 5728 . . . . . . . . . . . . 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 5719 . . . . . . . . . . . 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 5248 . . . . . . . . . . 11 |- ( f |` ( A \ ( `' f " ran E ) ) ) e. _V
58 cofunexg 6641 . . . . . . . . . . 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 3581 . . . . . . . . . . . . . 14 |- ( A \ ( `' f " ran E ) ) C_ A
6160, 8syl5sseqr 3503 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( A \ ( `' f " ran E ) ) C_ dom f )
62 fores 5727 . . . . . . . . . . . . 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 5574 . . . . . . . . . . . . . . . 16 |- ( Fun f -> Fun `' `' f )
65 imadif 5591 . . . . . . . . . . . . . . . 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 5267 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) )
68 difss 3581 . . . . . . . . . . . . . . 15 |- ( om \ ran E ) C_ om
69 foimacnv 5756 . . . . . . . . . . . . . . 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 5934 . . . . . . . . . . . . . . . . 17 |- ( f : A --> om -> ( `' f " om ) = A )
726, 71syl 16 . . . . . . . . . . . . . . . 16 |- ( f : A -onto-> om -> ( `' f " om ) = A )
7372difeq1d 3571 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( ( `' f " om ) \ ( `' f " ran E ) ) = ( A \ ( `' f " ran E ) ) )
7473imaeq2d 5267 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) = ( f " ( A \ ( `' f " ran E ) ) ) )
7567, 70, 743eqtr3rd 2501 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( A \ ( `' f " ran E ) ) ) = ( om \ ran E ) )
76 foeq3 5716 . . . . . . . . . . . . 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 5728 . . . . . . . . . . 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 7887 . . . . . . . . . 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 4538 . . . . . . . . . . 11 |- ( A e. _V -> ( A \ ( `' f " ran E ) ) e. _V )
84 isfin3-2 8637 . . . . . . . . . . 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 2523 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( y e. FinIII <-> ( `' f " ran E ) e. FinIII ) )
89 difeq2 3566 . . . . . . . . . . . . 13 |- ( y = ( `' f " ran E ) -> ( A \ y ) = ( A \ ( `' f " ran E ) ) )
9089eleq1d 2520 . . . . . . . . . . . 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 3169 . . . . . . . 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 1217 . . . . . . 7 |- ( f : A -onto-> om -> E. y e. ~P A -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
97 rexnal 2865 . . . . . . 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 1689 . . . . 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 8637 . . 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 1370   E.wex 1587   e. wcel 1758   =/= wne 2644   A.wral 2795   E.wrex 2796   _Vcvv 3068   \ cdif 3423   C_ wss 3426   (/)c0 3735   ~Pcpw 3958   class class class wbr 4390   |-> cmpt 4448   Oncon0 4817   suc csuc 4819   `'ccnv 4937   dom cdm 4938   ran crn 4939   |` cres 4940   "cima 4941   o. ccom 4942   Fun wfun 5510   -->wf 5512   -1-1->wf1 5513   -onto->wfo 5514   -1-1-onto->wf1o 5515  (class class class)co 6190   omcom 6576   2oc2o 7014   .o comu 7018   ~<_* cwdom 7873  FinIIIcfin3 8551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-seqom 7003  df-1o 7020  df-2o 7021  df-oadd 7024  df-omul 7025  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-wdom 7875  df-card 8210  df-fin4 8557  df-fin3 8558
This theorem is referenced by:  fin1a2lem8  8677
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