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

Theorem fin1a2lem7 8032
Description: Lemma for fin1a2 8041. 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 4675 . . . . . 6  |-  (/)  e.  om
2 ne0i 3461 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
3 brwdomn0 7283 . . . . . 6  |-  ( om  =/=  (/)  ->  ( om  ~<_*  A  <->  E. f  f : A -onto-> om ) )
41, 2, 3mp2b 9 . . . . 5  |-  ( om  ~<_*  A 
<->  E. f  f : A -onto-> om )
5 cnvimass 5033 . . . . . . . . . 10  |-  ( `' f " ran  E
)  C_  dom  f
6 fof 5451 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  f : A --> om )
7 fdm 5393 . . . . . . . . . . 11  |-  ( f : A --> om  ->  dom  f  =  A )
86, 7syl 15 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  dom  f  =  A )
95, 8syl5sseq 3226 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( `' f " ran  E )  C_  A )
10 vex 2791 . . . . . . . . . . 11  |-  f  e. 
_V
11 dmfex 5424 . . . . . . . . . . 11  |-  ( ( f  e.  _V  /\  f : A --> om )  ->  A  e.  _V )
1210, 6, 11sylancr 644 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  A  e.  _V )
13 elpw2g 4174 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
( `' f " ran  E )  e.  ~P A 
<->  ( `' f " ran  E )  C_  A
) )
1412, 13syl 15 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( ( `' f " ran  E )  e.  ~P A 
<->  ( `' f " ran  E )  C_  A
) )
159, 14mpbird 223 . . . . . . . 8  |-  ( f : A -onto-> om  ->  ( `' f " ran  E )  e.  ~P A
)
16 fin1a2lem.b . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
1716fin1a2lem4 8029 . . . . . . . . . . . . 13  |-  E : om
-1-1-> om
18 f1cnv 5497 . . . . . . . . . . . . 13  |-  ( E : om -1-1-> om  ->  `' E : ran  E -1-1-onto-> om )
19 f1ofo 5479 . . . . . . . . . . . . 13  |-  ( `' E : ran  E -1-1-onto-> om  ->  `' E : ran  E -onto-> om )
2017, 18, 19mp2b 9 . . . . . . . . . . . 12  |-  `' E : ran  E -onto-> om
21 fofun 5452 . . . . . . . . . . . 12  |-  ( `' E : ran  E -onto-> om  ->  Fun  `' E
)
2220, 21ax-mp 8 . . . . . . . . . . 11  |-  Fun  `' E
2310resex 4995 . . . . . . . . . . 11  |-  ( f  |`  ( `' f " ran  E ) )  e. 
_V
24 cofunexg 5739 . . . . . . . . . . 11  |-  ( ( Fun  `' E  /\  ( f  |`  ( `' f " ran  E ) )  e.  _V )  ->  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e. 
_V )
2522, 23, 24mp2an 653 . . . . . . . . . 10  |-  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e.  _V
26 fofun 5452 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  Fun  f )
27 fores 5460 . . . . . . . . . . . . 13  |-  ( ( Fun  f  /\  ( `' f " ran  E )  C_  dom  f )  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E ) -onto-> ( f "
( `' f " ran  E ) ) )
2826, 5, 27sylancl 643 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E
) -onto-> ( f "
( `' f " ran  E ) ) )
29 f1f 5437 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  E : om --> om )
30 frn 5395 . . . . . . . . . . . . . . 15  |-  ( E : om --> om  ->  ran 
E  C_  om )
3117, 29, 30mp2b 9 . . . . . . . . . . . . . 14  |-  ran  E  C_ 
om
32 foimacnv 5490 . . . . . . . . . . . . . 14  |-  ( ( f : A -onto-> om  /\ 
ran  E  C_  om )  ->  ( f " ( `' f " ran  E ) )  =  ran  E )
3331, 32mpan2 652 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ran  E
) )  =  ran  E )
34 foeq3 5449 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 201 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E
) -onto-> ran  E )
37 foco 5461 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( `' E  o.  (
f  |`  ( `' f
" ran  E )
) ) : ( `' f " ran  E ) -onto-> om )
39 fowdom 7285 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( `' f " ran  E ) )
41 cnvexg 5208 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  `' f  e.  _V )
42 imaexg 5026 . . . . . . . . . . . 12  |-  ( `' f  e.  _V  ->  ( `' f " ran  E )  e.  _V )
4310, 41, 42mp2b 9 . . . . . . . . . . 11  |-  ( `' f " ran  E
)  e.  _V
44 isfin3-2 7993 . . . . . . . . . . 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 322 . . . . . . . . 9  |-  ( om  ~<_*  ( `' f " ran  E )  <->  -.  ( `' f " ran  E )  e. FinIII )
4740, 46sylib 188 . . . . . . . 8  |-  ( f : A -onto-> om  ->  -.  ( `' f " ran  E )  e. FinIII )
48 fin1a2lem.aa . . . . . . . . . . . . . . 15  |-  S  =  ( x  e.  On  |->  suc  x )
4916, 48fin1a2lem6 8031 . . . . . . . . . . . . . 14  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
50 f1ocnv 5485 . . . . . . . . . . . . . 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 5479 . . . . . . . . . . . . . 14  |-  ( `' ( S  |`  ran  E
) : ( om 
\  ran  E ) -1-1-onto-> ran  E  ->  `' ( S  |`  ran  E ) : ( om  \  ran  E ) -onto-> ran  E )
5249, 50, 51mp2b 9 . . . . . . . . . . . . 13  |-  `' ( S  |`  ran  E ) : ( om  \  ran  E ) -onto-> ran  E
53 foco 5461 . . . . . . . . . . . . 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 653 . . . . . . . . . . . 12  |-  ( `' E  o.  `' ( S  |`  ran  E ) ) : ( om 
\  ran  E ) -onto-> om
55 fofun 5452 . . . . . . . . . . . 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 4995 . . . . . . . . . . 11  |-  ( f  |`  ( A  \  ( `' f " ran  E ) ) )  e. 
_V
58 cofunexg 5739 . . . . . . . . . . 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 653 . . . . . . . . . 10  |-  ( ( `' E  o.  `' ( S  |`  ran  E
) )  o.  (
f  |`  ( A  \ 
( `' f " ran  E ) ) ) )  e.  _V
60 difss 3303 . . . . . . . . . . . . . 14  |-  ( A 
\  ( `' f
" ran  E )
)  C_  A
6160, 8syl5sseqr 3227 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( A  \  ( `' f " ran  E
) )  C_  dom  f )
62 fores 5460 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( A  \  ( `' f " ran  E ) ) ) : ( A  \ 
( `' f " ran  E ) ) -onto-> ( f " ( A 
\  ( `' f
" ran  E )
) ) )
64 funcnvcnv 5308 . . . . . . . . . . . . . . . 16  |-  ( Fun  f  ->  Fun  `' `' f )
65 imadif 5327 . . . . . . . . . . . . . . . 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 5012 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) ) )
68 difss 3303 . . . . . . . . . . . . . . 15  |-  ( om 
\  ran  E )  C_ 
om
69 foimacnv 5490 . . . . . . . . . . . . . . 15  |-  ( ( f : A -onto-> om  /\  ( om  \  ran  E )  C_  om )  ->  ( f " ( `' f " ( om  \  ran  E ) ) )  =  ( om  \  ran  E
) )
7068, 69mpan2 652 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( om  \  ran  E
) )
71 fimacnv 5657 . . . . . . . . . . . . . . . . 17  |-  ( f : A --> om  ->  ( `' f " om )  =  A )
726, 71syl 15 . . . . . . . . . . . . . . . 16  |-  ( f : A -onto-> om  ->  ( `' f " om )  =  A )
7372difeq1d 3293 . . . . . . . . . . . . . . 15  |-  ( f : A -onto-> om  ->  ( ( `' f " om )  \  ( `' f " ran  E ) )  =  ( A  \  ( `' f " ran  E
) ) )
7473imaeq2d 5012 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) )  =  ( f " ( A 
\  ( `' f
" ran  E )
) ) )
7567, 70, 743eqtr3rd 2324 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( A 
\  ( `' f
" ran  E )
) )  =  ( om  \  ran  E
) )
76 foeq3 5449 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 201 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  ( f  |`  ( A  \  ( `' f " ran  E ) ) ) : ( A  \ 
( `' f " ran  E ) ) -onto-> ( om  \  ran  E
) )
79 foco 5461 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( ( `' E  o.  `' ( S  |`  ran  E ) )  o.  ( f  |`  ( A  \  ( `' f
" ran  E )
) ) ) : ( A  \  ( `' f " ran  E ) ) -onto-> om )
81 fowdom 7285 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( A  \  ( `' f " ran  E ) ) )
83 difexg 4162 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  \  ( `' f
" ran  E )
)  e.  _V )
84 isfin3-2 7993 . . . . . . . . . . 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 319 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( om  ~<_*  ( A  \  ( `' f " ran  E ) )  <->  -.  ( A  \  ( `' f
" ran  E )
)  e. FinIII ) )
8782, 86mpbid 201 . . . . . . . 8  |-  ( f : A -onto-> om  ->  -.  ( A  \  ( `' f " ran  E ) )  e. FinIII )
88 eleq1 2343 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( y  e. FinIII  <->  ( `' f " ran  E )  e. FinIII ) )
89 difeq2 3288 . . . . . . . . . . . . 13  |-  ( y  =  ( `' f
" ran  E )  ->  ( A  \  y
)  =  ( A 
\  ( `' f
" ran  E )
) )
9089eleq1d 2349 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( ( A  \ 
y )  e. FinIII  <->  ( A  \  ( `' f " ran  E ) )  e. FinIII ) )
9188, 90orbi12d 690 . . . . . . . . . . 11  |-  ( y  =  ( `' f
" ran  E )  ->  ( ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII )  <->  ( ( `' f " ran  E )  e. FinIII  \/  ( A  \  ( `' f " ran  E ) )  e. FinIII ) ) )
9291notbid 285 . . . . . . . . . 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 476 . . . . . . . . . 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 252 . . . . . . . . 9  |-  ( y  =  ( `' f
" ran  E )  ->  ( -.  ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII )  <->  ( -.  ( `' f " ran  E )  e. FinIII  /\  -.  ( A  \  ( `' f
" ran  E )
)  e. FinIII ) ) )
9594rspcev 2884 . . . . . . . 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 1180 . . . . . . 7  |-  ( f : A -onto-> om  ->  E. y  e.  ~P  A  -.  ( y  e. FinIII  \/  ( A  \  y )  e. FinIII ) )
97 rexnal 2554 . . . . . . 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 188 . . . . . 6  |-  ( f : A -onto-> om  ->  -. 
A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII ) )
9998exlimiv 1666 . . . . 5  |-  ( E. f  f : A -onto-> om  ->  -.  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII ) )
1004, 99sylbi 187 . . . 4  |-  ( om  ~<_*  A  ->  -.  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII ) )
101100con2i 112 . . 3  |-  ( A. y  e.  ~P  A
( y  e. FinIII  \/  ( A  \  y )  e. FinIII )  ->  -.  om  ~<_*  A )
102 isfin3-2 7993 . . 3  |-  ( A  e.  V  ->  ( A  e. FinIII 
<->  -.  om  ~<_*  A ) )
103101, 102syl5ibr 212 . 2  |-  ( A  e.  V  ->  ( A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII )  ->  A  e. FinIII ) )
104103imp 418 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 176    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   Oncon0 4392   suc csuc 4394   omcom 4656   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254  (class class class)co 5858   2oc2o 6473    .o comu 6477    ~<_* cwdom 7271  FinIIIcfin3 7907
This theorem is referenced by:  fin1a2lem8  8033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-wdom 7273  df-card 7572  df-fin4 7913  df-fin3 7914
  Copyright terms: Public domain W3C validator