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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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