MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1a2lem7 Structured version   Visualization version   Unicode version

Theorem fin1a2lem7 8854
Description: Lemma for fin1a2 8863. 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 6731 . . . . . 6  |-  (/)  e.  om
2 ne0i 3728 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
3 brwdomn0 8102 . . . . . 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 5194 . . . . . . . . . 10  |-  ( `' f " ran  E
)  C_  dom  f
6 fof 5806 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  f : A --> om )
7 fdm 5745 . . . . . . . . . . 11  |-  ( f : A --> om  ->  dom  f  =  A )
86, 7syl 17 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  dom  f  =  A )
95, 8syl5sseq 3466 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( `' f " ran  E )  C_  A )
10 vex 3034 . . . . . . . . . . 11  |-  f  e. 
_V
11 dmfex 6770 . . . . . . . . . . 11  |-  ( ( f  e.  _V  /\  f : A --> om )  ->  A  e.  _V )
1210, 6, 11sylancr 676 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  A  e.  _V )
13 elpw2g 4564 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
( `' f " ran  E )  e.  ~P A 
<->  ( `' f " ran  E )  C_  A
) )
1412, 13syl 17 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( ( `' f " ran  E )  e.  ~P A 
<->  ( `' f " ran  E )  C_  A
) )
159, 14mpbird 240 . . . . . . . 8  |-  ( f : A -onto-> om  ->  ( `' f " ran  E )  e.  ~P A
)
16 fin1a2lem.b . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
1716fin1a2lem4 8851 . . . . . . . . . . . . 13  |-  E : om
-1-1-> om
18 f1cnv 5851 . . . . . . . . . . . . 13  |-  ( E : om -1-1-> om  ->  `' E : ran  E -1-1-onto-> om )
19 f1ofo 5835 . . . . . . . . . . . . 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 5807 . . . . . . . . . . . 12  |-  ( `' E : ran  E -onto-> om  ->  Fun  `' E
)
2220, 21ax-mp 5 . . . . . . . . . . 11  |-  Fun  `' E
2310resex 5154 . . . . . . . . . . 11  |-  ( f  |`  ( `' f " ran  E ) )  e. 
_V
24 cofunexg 6776 . . . . . . . . . . 11  |-  ( ( Fun  `' E  /\  ( f  |`  ( `' f " ran  E ) )  e.  _V )  ->  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e. 
_V )
2522, 23, 24mp2an 686 . . . . . . . . . 10  |-  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e.  _V
26 fofun 5807 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  Fun  f )
27 fores 5815 . . . . . . . . . . . . 13  |-  ( ( Fun  f  /\  ( `' f " ran  E )  C_  dom  f )  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E ) -onto-> ( f "
( `' f " ran  E ) ) )
2826, 5, 27sylancl 675 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E
) -onto-> ( f "
( `' f " ran  E ) ) )
29 f1f 5792 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  E : om --> om )
30 frn 5747 . . . . . . . . . . . . . . 15  |-  ( E : om --> om  ->  ran 
E  C_  om )
3117, 29, 30mp2b 10 . . . . . . . . . . . . . 14  |-  ran  E  C_ 
om
32 foimacnv 5845 . . . . . . . . . . . . . 14  |-  ( ( f : A -onto-> om  /\ 
ran  E  C_  om )  ->  ( f " ( `' f " ran  E ) )  =  ran  E )
3331, 32mpan2 685 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ran  E
) )  =  ran  E )
34 foeq3 5804 . . . . . . . . . . . . 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 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
) )
3628, 35mpbid 215 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E
) -onto-> ran  E )
37 foco 5816 . . . . . . . . . . 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 676 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( `' E  o.  (
f  |`  ( `' f
" ran  E )
) ) : ( `' f " ran  E ) -onto-> om )
39 fowdom 8104 . . . . . . . . . 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 676 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( `' f " ran  E ) )
41 cnvexg 6758 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  `' f  e.  _V )
42 imaexg 6749 . . . . . . . . . . . 12  |-  ( `' f  e.  _V  ->  ( `' f " ran  E )  e.  _V )
4310, 41, 42mp2b 10 . . . . . . . . . . 11  |-  ( `' f " ran  E
)  e.  _V
44 isfin3-2 8815 . . . . . . . . . . 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 339 . . . . . . . . 9  |-  ( om  ~<_*  ( `' f " ran  E )  <->  -.  ( `' f " ran  E )  e. FinIII )
4740, 46sylib 201 . . . . . . . 8  |-  ( f : A -onto-> om  ->  -.  ( `' f " ran  E )  e. FinIII )
48 fin1a2lem.aa . . . . . . . . . . . . . . 15  |-  S  =  ( x  e.  On  |->  suc  x )
4916, 48fin1a2lem6 8853 . . . . . . . . . . . . . 14  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
50 f1ocnv 5840 . . . . . . . . . . . . . 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 5835 . . . . . . . . . . . . . 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 5816 . . . . . . . . . . . . 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 686 . . . . . . . . . . . 12  |-  ( `' E  o.  `' ( S  |`  ran  E ) ) : ( om 
\  ran  E ) -onto-> om
55 fofun 5807 . . . . . . . . . . . 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 5154 . . . . . . . . . . 11  |-  ( f  |`  ( A  \  ( `' f " ran  E ) ) )  e. 
_V
58 cofunexg 6776 . . . . . . . . . . 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 686 . . . . . . . . . 10  |-  ( ( `' E  o.  `' ( S  |`  ran  E
) )  o.  (
f  |`  ( A  \ 
( `' f " ran  E ) ) ) )  e.  _V
60 difss 3549 . . . . . . . . . . . . . 14  |-  ( A 
\  ( `' f
" ran  E )
)  C_  A
6160, 8syl5sseqr 3467 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( A  \  ( `' f " ran  E
) )  C_  dom  f )
62 fores 5815 . . . . . . . . . . . . 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 673 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( A  \  ( `' f " ran  E ) ) ) : ( A  \ 
( `' f " ran  E ) ) -onto-> ( f " ( A 
\  ( `' f
" ran  E )
) ) )
64 funcnvcnv 5651 . . . . . . . . . . . . . . . 16  |-  ( Fun  f  ->  Fun  `' `' f )
65 imadif 5668 . . . . . . . . . . . . . . . 16  |-  ( Fun  `' `' f  ->  ( `' f " ( om 
\  ran  E )
)  =  ( ( `' f " om )  \  ( `' f
" ran  E )
) )
6626, 64, 653syl 18 . . . . . . . . . . . . . . 15  |-  ( f : A -onto-> om  ->  ( `' f " ( om  \  ran  E ) )  =  ( ( `' f " om )  \  ( `' f
" ran  E )
) )
6766imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) ) )
68 difss 3549 . . . . . . . . . . . . . . 15  |-  ( om 
\  ran  E )  C_ 
om
69 foimacnv 5845 . . . . . . . . . . . . . . 15  |-  ( ( f : A -onto-> om  /\  ( om  \  ran  E )  C_  om )  ->  ( f " ( `' f " ( om  \  ran  E ) ) )  =  ( om  \  ran  E
) )
7068, 69mpan2 685 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( om  \  ran  E
) )
71 fimacnv 6027 . . . . . . . . . . . . . . . . 17  |-  ( f : A --> om  ->  ( `' f " om )  =  A )
726, 71syl 17 . . . . . . . . . . . . . . . 16  |-  ( f : A -onto-> om  ->  ( `' f " om )  =  A )
7372difeq1d 3539 . . . . . . . . . . . . . . 15  |-  ( f : A -onto-> om  ->  ( ( `' f " om )  \  ( `' f " ran  E ) )  =  ( A  \  ( `' f " ran  E
) ) )
7473imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) )  =  ( f " ( A 
\  ( `' f
" ran  E )
) ) )
7567, 70, 743eqtr3rd 2514 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( A 
\  ( `' f
" ran  E )
) )  =  ( om  \  ran  E
) )
76 foeq3 5804 . . . . . . . . . . . . 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 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 )
) )
7863, 77mpbid 215 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  ( f  |`  ( A  \  ( `' f " ran  E ) ) ) : ( A  \ 
( `' f " ran  E ) ) -onto-> ( om  \  ran  E
) )
79 foco 5816 . . . . . . . . . . 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 676 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( ( `' E  o.  `' ( S  |`  ran  E ) )  o.  ( f  |`  ( A  \  ( `' f
" ran  E )
) ) ) : ( A  \  ( `' f " ran  E ) ) -onto-> om )
81 fowdom 8104 . . . . . . . . . 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 676 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( A  \  ( `' f " ran  E ) ) )
83 difexg 4545 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  \  ( `' f
" ran  E )
)  e.  _V )
84 isfin3-2 8815 . . . . . . . . . . 11  |-  ( ( A  \  ( `' f " ran  E
) )  e.  _V  ->  ( ( A  \ 
( `' f " ran  E ) )  e. FinIII  <->  -.  om  ~<_*  ( A  \  ( `' f " ran  E ) ) ) )
8512, 83, 843syl 18 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( ( A  \  ( `' f " ran  E ) )  e. FinIII  <->  -.  om  ~<_*  ( A  \  ( `' f " ran  E ) ) ) )
8685con2bid 336 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( om  ~<_*  ( A  \  ( `' f " ran  E ) )  <->  -.  ( A  \  ( `' f
" ran  E )
)  e. FinIII ) )
8782, 86mpbid 215 . . . . . . . 8  |-  ( f : A -onto-> om  ->  -.  ( A  \  ( `' f " ran  E ) )  e. FinIII )
88 eleq1 2537 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( y  e. FinIII  <->  ( `' f " ran  E )  e. FinIII ) )
89 difeq2 3534 . . . . . . . . . . . . 13  |-  ( y  =  ( `' f
" ran  E )  ->  ( A  \  y
)  =  ( A 
\  ( `' f
" ran  E )
) )
9089eleq1d 2533 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( ( A  \ 
y )  e. FinIII  <->  ( A  \  ( `' f " ran  E ) )  e. FinIII ) )
9188, 90orbi12d 724 . . . . . . . . . . 11  |-  ( y  =  ( `' f
" ran  E )  ->  ( ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII )  <->  ( ( `' f " ran  E )  e. FinIII  \/  ( A  \  ( `' f " ran  E ) )  e. FinIII ) ) )
9291notbid 301 . . . . . . . . . 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 498 . . . . . . . . . 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 269 . . . . . . . . 9  |-  ( y  =  ( `' f
" ran  E )  ->  ( -.  ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII )  <->  ( -.  ( `' f " ran  E )  e. FinIII  /\  -.  ( A  \  ( `' f
" ran  E )
)  e. FinIII ) ) )
9594rspcev 3136 . . . . . . . 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 1290 . . . . . . 7  |-  ( f : A -onto-> om  ->  E. y  e.  ~P  A  -.  ( y  e. FinIII  \/  ( A  \  y )  e. FinIII ) )
97 rexnal 2836 . . . . . . 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 201 . . . . . 6  |-  ( f : A -onto-> om  ->  -. 
A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII ) )
9998exlimiv 1784 . . . . 5  |-  ( E. f  f : A -onto-> om  ->  -.  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII ) )
1004, 99sylbi 200 . . . 4  |-  ( om  ~<_*  A  ->  -.  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII ) )
101100con2i 124 . . 3  |-  ( A. y  e.  ~P  A
( y  e. FinIII  \/  ( A  \  y )  e. FinIII )  ->  -.  om  ~<_*  A )
102 isfin3-2 8815 . . 3  |-  ( A  e.  V  ->  ( A  e. FinIII 
<->  -.  om  ~<_*  A ) )
103101, 102syl5ibr 229 . 2  |-  ( A  e.  V  ->  ( A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII )  ->  A  e. FinIII ) )
104103imp 436 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 189    \/ wo 375    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   class class class wbr 4395    |-> cmpt 4454   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842    o. ccom 4843   Oncon0 5430   suc csuc 5432   Fun wfun 5583   -->wf 5585   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588  (class class class)co 6308   omcom 6711   2oc2o 7194    .o comu 7198    ~<_* cwdom 8090  FinIIIcfin3 8729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-seqom 7183  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-wdom 8092  df-card 8391  df-fin4 8735  df-fin3 8736
This theorem is referenced by:  fin1a2lem8  8855
  Copyright terms: Public domain W3C validator