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Theorem fin1a2lem7 9228
Description: Lemma for fin1a2 9237. 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 7085 . . . . . 6 |- (/) e. om
2 ne0i 3921 . . . . . 6 |- ( (/) e. om -> om =/= (/) )
3 brwdomn0 8474 . . . . . 6 |- ( om =/= (/) -> ( om ~<_* A <-> E. f f : A -onto-> om ) )
41, 2, 3mp2b 10 . . . . 5 |- ( om ~<_* A <-> E. f f : A -onto-> om )
5 vex 3203 . . . . . . . . . 10 |- f e. _V
6 fof 6115 . . . . . . . . . 10 |- ( f : A -onto-> om -> f : A --> om )
7 dmfex 7124 . . . . . . . . . 10 |- ( ( f e. _V /\ f : A --> om ) -> A e. _V )
85, 6, 7sylancr 695 . . . . . . . . 9 |- ( f : A -onto-> om -> A e. _V )
9 cnvimass 5485 . . . . . . . . . 10 |- ( `' f " ran E ) C_ dom f
10 fdm 6051 . . . . . . . . . . 11 |- ( f : A --> om -> dom f = A )
116, 10syl 17 . . . . . . . . . 10 |- ( f : A -onto-> om -> dom f = A )
129, 11syl5sseq 3653 . . . . . . . . 9 |- ( f : A -onto-> om -> ( `' f " ran E ) C_ A )
138, 12sselpwd 4807 . . . . . . . 8 |- ( f : A -onto-> om -> ( `' f " ran E ) e. ~P A )
14 fin1a2lem.b . . . . . . . . . . . . . 14 |- E = ( x e. om |-> ( 2o .o x ) )
1514fin1a2lem4 9225 . . . . . . . . . . . . 13 |- E : om -1-1-> om
16 f1cnv 6160 . . . . . . . . . . . . 13 |- ( E : om -1-1-> om -> `' E : ran E -1-1-onto-> om )
17 f1ofo 6144 . . . . . . . . . . . . 13 |- ( `' E : ran E -1-1-onto-> om -> `' E : ran E -onto-> om )
1815, 16, 17mp2b 10 . . . . . . . . . . . 12 |- `' E : ran E -onto-> om
19 fofun 6116 . . . . . . . . . . . 12 |- ( `' E : ran E -onto-> om -> Fun `' E )
2018, 19ax-mp 5 . . . . . . . . . . 11 |- Fun `' E
215resex 5443 . . . . . . . . . . 11 |- ( f |` ( `' f " ran E ) ) e. _V
22 cofunexg 7130 . . . . . . . . . . 11 |- ( ( Fun `' E /\ ( f |` ( `' f " ran E ) ) e. _V ) -> ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V )
2320, 21, 22mp2an 708 . . . . . . . . . 10 |- ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V
24 fofun 6116 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> Fun f )
25 fores 6124 . . . . . . . . . . . . 13 |- ( ( Fun f /\ ( `' f " ran E ) C_ dom f ) -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
2624, 9, 25sylancl 694 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ( f " ( `' f " ran E ) ) )
27 f1f 6101 . . . . . . . . . . . . . . 15 |- ( E : om -1-1-> om -> E : om --> om )
28 frn 6053 . . . . . . . . . . . . . . 15 |- ( E : om --> om -> ran E C_ om )
2915, 27, 28mp2b 10 . . . . . . . . . . . . . 14 |- ran E C_ om
30 foimacnv 6154 . . . . . . . . . . . . . 14 |- ( ( f : A -onto-> om /\ ran E C_ om ) -> ( f " ( `' f " ran E ) ) = ran E )
3129, 30mpan2 707 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( `' f " ran E ) ) = ran E )
32 foeq3 6113 . . . . . . . . . . . . 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 ) )
3331, 32syl 17 . . . . . . . . . . . 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 ) )
3426, 33mpbid 222 . . . . . . . . . . 11 |- ( f : A -onto-> om -> ( f |` ( `' f " ran E ) ) : ( `' f " ran E ) -onto-> ran E )
35 foco 6125 . . . . . . . . . . 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 )
3618, 34, 35sylancr 695 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( `' E o. ( f |` ( `' f " ran E ) ) ) : ( `' f " ran E ) -onto-> om )
37 fowdom 8476 . . . . . . . . . 10 |- ( ( ( `' E o. ( f |` ( `' f " ran E ) ) ) e. _V /\ ( `' E o. ( f |` ( `' f " ran E ) ) ) : ( `' f " ran E ) -onto-> om ) -> om ~<_* ( `' f " ran E ) )
3823, 36, 37sylancr 695 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( `' f " ran E ) )
395cnvex 7113 . . . . . . . . . . . 12 |- `' f e. _V
4039imaex 7104 . . . . . . . . . . 11 |- ( `' f " ran E ) e. _V
41 isfin3-2 9189 . . . . . . . . . . 11 |- ( ( `' f " ran E ) e. _V -> ( ( `' f " ran E ) e. FinIII <-> -. om ~<_* ( `' f " ran E ) ) )
4240, 41ax-mp 5 . . . . . . . . . 10 |- ( ( `' f " ran E ) e. FinIII <-> -. om ~<_* ( `' f " ran E ) )
4342con2bii 347 . . . . . . . . 9 |- ( om ~<_* ( `' f " ran E ) <-> -. ( `' f " ran E ) e. FinIII )
4438, 43sylib 208 . . . . . . . 8 |- ( f : A -onto-> om -> -. ( `' f " ran E ) e. FinIII )
45 fin1a2lem.aa . . . . . . . . . . . . . . 15 |- S = ( x e. On |-> suc x )
4614, 45fin1a2lem6 9227 . . . . . . . . . . . . . 14 |- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )
47 f1ocnv 6149 . . . . . . . . . . . . . 14 |- ( ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E ) -> `' ( S |` ran E ) : ( om \ ran E ) -1-1-onto-> ran E )
48 f1ofo 6144 . . . . . . . . . . . . . 14 |- ( `' ( S |` ran E ) : ( om \ ran E ) -1-1-onto-> ran E -> `' ( S |` ran E ) : ( om \ ran E ) -onto-> ran E )
4946, 47, 48mp2b 10 . . . . . . . . . . . . 13 |- `' ( S |` ran E ) : ( om \ ran E ) -onto-> ran E
50 foco 6125 . . . . . . . . . . . . 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 )
5118, 49, 50mp2an 708 . . . . . . . . . . . 12 |- ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om
52 fofun 6116 . . . . . . . . . . . 12 |- ( ( `' E o. `' ( S |` ran E ) ) : ( om \ ran E ) -onto-> om -> Fun ( `' E o. `' ( S |` ran E ) ) )
5351, 52ax-mp 5 . . . . . . . . . . 11 |- Fun ( `' E o. `' ( S |` ran E ) )
545resex 5443 . . . . . . . . . . 11 |- ( f |` ( A \ ( `' f " ran E ) ) ) e. _V
55 cofunexg 7130 . . . . . . . . . . 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 )
5653, 54, 55mp2an 708 . . . . . . . . . 10 |- ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) e. _V
57 difss 3737 . . . . . . . . . . . . . 14 |- ( A \ ( `' f " ran E ) ) C_ A
5857, 11syl5sseqr 3654 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( A \ ( `' f " ran E ) ) C_ dom f )
59 fores 6124 . . . . . . . . . . . . 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 ) ) ) )
6024, 58, 59syl2anc 693 . . . . . . . . . . . 12 |- ( f : A -onto-> om -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( f " ( A \ ( `' f " ran E ) ) ) )
61 funcnvcnv 5956 . . . . . . . . . . . . . . . 16 |- ( Fun f -> Fun `' `' f )
62 imadif 5973 . . . . . . . . . . . . . . . 16 |- ( Fun `' `' f -> ( `' f " ( om \ ran E ) ) = ( ( `' f " om ) \ ( `' f " ran E ) ) )
6324, 61, 623syl 18 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( `' f " ( om \ ran E ) ) = ( ( `' f " om ) \ ( `' f " ran E ) ) )
6463imaeq2d 5466 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) )
65 difss 3737 . . . . . . . . . . . . . . 15 |- ( om \ ran E ) C_ om
66 foimacnv 6154 . . . . . . . . . . . . . . 15 |- ( ( f : A -onto-> om /\ ( om \ ran E ) C_ om ) -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
6765, 66mpan2 707 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( `' f " ( om \ ran E ) ) ) = ( om \ ran E ) )
68 fimacnv 6347 . . . . . . . . . . . . . . . . 17 |- ( f : A --> om -> ( `' f " om ) = A )
696, 68syl 17 . . . . . . . . . . . . . . . 16 |- ( f : A -onto-> om -> ( `' f " om ) = A )
7069difeq1d 3727 . . . . . . . . . . . . . . 15 |- ( f : A -onto-> om -> ( ( `' f " om ) \ ( `' f " ran E ) ) = ( A \ ( `' f " ran E ) ) )
7170imaeq2d 5466 . . . . . . . . . . . . . 14 |- ( f : A -onto-> om -> ( f " ( ( `' f " om ) \ ( `' f " ran E ) ) ) = ( f " ( A \ ( `' f " ran E ) ) ) )
7264, 67, 713eqtr3rd 2665 . . . . . . . . . . . . 13 |- ( f : A -onto-> om -> ( f " ( A \ ( `' f " ran E ) ) ) = ( om \ ran E ) )
73 foeq3 6113 . . . . . . . . . . . . 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 ) ) )
7472, 73syl 17 . . . . . . . . . . . 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 ) ) )
7560, 74mpbid 222 . . . . . . . . . . 11 |- ( f : A -onto-> om -> ( f |` ( A \ ( `' f " ran E ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> ( om \ ran E ) )
76 foco 6125 . . . . . . . . . . 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 )
7751, 75, 76sylancr 695 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( ( `' E o. `' ( S |` ran E ) ) o. ( f |` ( A \ ( `' f " ran E ) ) ) ) : ( A \ ( `' f " ran E ) ) -onto-> om )
78 fowdom 8476 . . . . . . . . . 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 ) ) )
7956, 77, 78sylancr 695 . . . . . . . . 9 |- ( f : A -onto-> om -> om ~<_* ( A \ ( `' f " ran E ) ) )
80 difexg 4808 . . . . . . . . . . 11 |- ( A e. _V -> ( A \ ( `' f " ran E ) ) e. _V )
81 isfin3-2 9189 . . . . . . . . . . 11 |- ( ( A \ ( `' f " ran E ) ) e. _V -> ( ( A \ ( `' f " ran E ) ) e. FinIII <-> -. om ~<_* ( A \ ( `' f " ran E ) ) ) )
828, 80, 813syl 18 . . . . . . . . . 10 |- ( f : A -onto-> om -> ( ( A \ ( `' f " ran E ) ) e. FinIII <-> -. om ~<_* ( A \ ( `' f " ran E ) ) ) )
8382con2bid 344 . . . . . . . . 9 |- ( f : A -onto-> om -> ( om ~<_* ( A \ ( `' f " ran E ) ) <-> -. ( A \ ( `' f " ran E ) ) e. FinIII ) )
8479, 83mpbid 222 . . . . . . . 8 |- ( f : A -onto-> om -> -. ( A \ ( `' f " ran E ) ) e. FinIII )
85 eleq1 2689 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( y e. FinIII <-> ( `' f " ran E ) e. FinIII ) )
86 difeq2 3722 . . . . . . . . . . . . 13 |- ( y = ( `' f " ran E ) -> ( A \ y ) = ( A \ ( `' f " ran E ) ) )
8786eleq1d 2686 . . . . . . . . . . . 12 |- ( y = ( `' f " ran E ) -> ( ( A \ y ) e. FinIII <-> ( A \ ( `' f " ran E ) ) e. FinIII ) )
8885, 87orbi12d 746 . . . . . . . . . . 11 |- ( y = ( `' f " ran E ) -> ( ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> ( ( `' f " ran E ) e. FinIII \/ ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
8988notbid 308 . . . . . . . . . 10 |- ( y = ( `' f " ran E ) -> ( -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> -. ( ( `' f " ran E ) e. FinIII \/ ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
90 ioran 511 . . . . . . . . . 10 |- ( -. ( ( `' f " ran E ) e. FinIII \/ ( A \ ( `' f " ran E ) ) e. FinIII ) <-> ( -. ( `' f " ran E ) e. FinIII /\ -. ( A \ ( `' f " ran E ) ) e. FinIII ) )
9189, 90syl6bb 276 . . . . . . . . 9 |- ( y = ( `' f " ran E ) -> ( -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) <-> ( -. ( `' f " ran E ) e. FinIII /\ -. ( A \ ( `' f " ran E ) ) e. FinIII ) ) )
9291rspcev 3309 . . . . . . . 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 ) )
9313, 44, 84, 92syl12anc 1324 . . . . . . 7 |- ( f : A -onto-> om -> E. y e. ~P A -. ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
94 rexnal 2995 . . . . . . 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 ) )
9593, 94sylib 208 . . . . . 6 |- ( f : A -onto-> om -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
9695exlimiv 1858 . . . . 5 |- ( E. f f : A -onto-> om -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
974, 96sylbi 207 . . . 4 |- ( om ~<_* A -> -. A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) )
9897con2i 134 . . 3 |- ( A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) -> -. om ~<_* A )
99 isfin3-2 9189 . . 3 |- ( A e. V -> ( A e. FinIII <-> -. om ~<_* A ) )
10098, 99syl5ibr 236 . 2 |- ( A e. V -> ( A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) -> A e. FinIII ) )
101100imp 445 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 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Oncon0 5723   suc csuc 5725   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887  (class class class)co 6650   omcom 7065   2oc2o 7554   .o comu 7558   ~<_* cwdom 8462  FinIIIcfin3 9103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wdom 8464  df-card 8765  df-fin4 9109  df-fin3 9110
This theorem is referenced by:  fin1a2lem8  9229
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