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Theorem fin1a2lem7 8286
Description: Lemma for fin1a2 8295. 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 4864 . . . . . 6  |-  (/)  e.  om
2 ne0i 3634 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
3 brwdomn0 7537 . . . . . 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 5224 . . . . . . . . . 10  |-  ( `' f " ran  E
)  C_  dom  f
6 fof 5653 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  f : A --> om )
7 fdm 5595 . . . . . . . . . . 11  |-  ( f : A --> om  ->  dom  f  =  A )
86, 7syl 16 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  dom  f  =  A )
95, 8syl5sseq 3396 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( `' f " ran  E )  C_  A )
10 vex 2959 . . . . . . . . . . 11  |-  f  e. 
_V
11 dmfex 5626 . . . . . . . . . . 11  |-  ( ( f  e.  _V  /\  f : A --> om )  ->  A  e.  _V )
1210, 6, 11sylancr 645 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  A  e.  _V )
13 elpw2g 4363 . . . . . . . . . 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 224 . . . . . . . 8  |-  ( f : A -onto-> om  ->  ( `' f " ran  E )  e.  ~P A
)
16 fin1a2lem.b . . . . . . . . . . . . . 14  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
1716fin1a2lem4 8283 . . . . . . . . . . . . 13  |-  E : om
-1-1-> om
18 f1cnv 5699 . . . . . . . . . . . . 13  |-  ( E : om -1-1-> om  ->  `' E : ran  E -1-1-onto-> om )
19 f1ofo 5681 . . . . . . . . . . . . 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 5654 . . . . . . . . . . . 12  |-  ( `' E : ran  E -onto-> om  ->  Fun  `' E
)
2220, 21ax-mp 8 . . . . . . . . . . 11  |-  Fun  `' E
2310resex 5186 . . . . . . . . . . 11  |-  ( f  |`  ( `' f " ran  E ) )  e. 
_V
24 cofunexg 5959 . . . . . . . . . . 11  |-  ( ( Fun  `' E  /\  ( f  |`  ( `' f " ran  E ) )  e.  _V )  ->  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e. 
_V )
2522, 23, 24mp2an 654 . . . . . . . . . 10  |-  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e.  _V
26 fofun 5654 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  Fun  f )
27 fores 5662 . . . . . . . . . . . . 13  |-  ( ( Fun  f  /\  ( `' f " ran  E )  C_  dom  f )  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E ) -onto-> ( f "
( `' f " ran  E ) ) )
2826, 5, 27sylancl 644 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E
) -onto-> ( f "
( `' f " ran  E ) ) )
29 f1f 5639 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  E : om --> om )
30 frn 5597 . . . . . . . . . . . . . . 15  |-  ( E : om --> om  ->  ran 
E  C_  om )
3117, 29, 30mp2b 10 . . . . . . . . . . . . . 14  |-  ran  E  C_ 
om
32 foimacnv 5692 . . . . . . . . . . . . . 14  |-  ( ( f : A -onto-> om  /\ 
ran  E  C_  om )  ->  ( f " ( `' f " ran  E ) )  =  ran  E )
3331, 32mpan2 653 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ran  E
) )  =  ran  E )
34 foeq3 5651 . . . . . . . . . . . . 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 202 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E
) -onto-> ran  E )
37 foco 5663 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( `' E  o.  (
f  |`  ( `' f
" ran  E )
) ) : ( `' f " ran  E ) -onto-> om )
39 fowdom 7539 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( `' f " ran  E ) )
41 cnvexg 5405 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  `' f  e.  _V )
42 imaexg 5217 . . . . . . . . . . . 12  |-  ( `' f  e.  _V  ->  ( `' f " ran  E )  e.  _V )
4310, 41, 42mp2b 10 . . . . . . . . . . 11  |-  ( `' f " ran  E
)  e.  _V
44 isfin3-2 8247 . . . . . . . . . . 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 323 . . . . . . . . 9  |-  ( om  ~<_*  ( `' f " ran  E )  <->  -.  ( `' f " ran  E )  e. FinIII )
4740, 46sylib 189 . . . . . . . 8  |-  ( f : A -onto-> om  ->  -.  ( `' f " ran  E )  e. FinIII )
48 fin1a2lem.aa . . . . . . . . . . . . . . 15  |-  S  =  ( x  e.  On  |->  suc  x )
4916, 48fin1a2lem6 8285 . . . . . . . . . . . . . 14  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
50 f1ocnv 5687 . . . . . . . . . . . . . 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 5681 . . . . . . . . . . . . . 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 5663 . . . . . . . . . . . . 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 654 . . . . . . . . . . . 12  |-  ( `' E  o.  `' ( S  |`  ran  E ) ) : ( om 
\  ran  E ) -onto-> om
55 fofun 5654 . . . . . . . . . . . 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 5186 . . . . . . . . . . 11  |-  ( f  |`  ( A  \  ( `' f " ran  E ) ) )  e. 
_V
58 cofunexg 5959 . . . . . . . . . . 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 654 . . . . . . . . . 10  |-  ( ( `' E  o.  `' ( S  |`  ran  E
) )  o.  (
f  |`  ( A  \ 
( `' f " ran  E ) ) ) )  e.  _V
60 difss 3474 . . . . . . . . . . . . . 14  |-  ( A 
\  ( `' f
" ran  E )
)  C_  A
6160, 8syl5sseqr 3397 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( A  \  ( `' f " ran  E
) )  C_  dom  f )
62 fores 5662 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( A  \  ( `' f " ran  E ) ) ) : ( A  \ 
( `' f " ran  E ) ) -onto-> ( f " ( A 
\  ( `' f
" ran  E )
) ) )
64 funcnvcnv 5509 . . . . . . . . . . . . . . . 16  |-  ( Fun  f  ->  Fun  `' `' f )
65 imadif 5528 . . . . . . . . . . . . . . . 16  |-  ( Fun  `' `' f  ->  ( `' f " ( om 
\  ran  E )
)  =  ( ( `' f " om )  \  ( `' f
" ran  E )
) )
6626, 64, 653syl 19 . . . . . . . . . . . . . . 15  |-  ( f : A -onto-> om  ->  ( `' f " ( om  \  ran  E ) )  =  ( ( `' f " om )  \  ( `' f
" ran  E )
) )
6766imaeq2d 5203 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) ) )
68 difss 3474 . . . . . . . . . . . . . . 15  |-  ( om 
\  ran  E )  C_ 
om
69 foimacnv 5692 . . . . . . . . . . . . . . 15  |-  ( ( f : A -onto-> om  /\  ( om  \  ran  E )  C_  om )  ->  ( f " ( `' f " ( om  \  ran  E ) ) )  =  ( om  \  ran  E
) )
7068, 69mpan2 653 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( om  \  ran  E
) )
71 fimacnv 5862 . . . . . . . . . . . . . . . . 17  |-  ( f : A --> om  ->  ( `' f " om )  =  A )
726, 71syl 16 . . . . . . . . . . . . . . . 16  |-  ( f : A -onto-> om  ->  ( `' f " om )  =  A )
7372difeq1d 3464 . . . . . . . . . . . . . . 15  |-  ( f : A -onto-> om  ->  ( ( `' f " om )  \  ( `' f " ran  E ) )  =  ( A  \  ( `' f " ran  E
) ) )
7473imaeq2d 5203 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) )  =  ( f " ( A 
\  ( `' f
" ran  E )
) ) )
7567, 70, 743eqtr3rd 2477 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( A 
\  ( `' f
" ran  E )
) )  =  ( om  \  ran  E
) )
76 foeq3 5651 . . . . . . . . . . . . 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 202 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  ( f  |`  ( A  \  ( `' f " ran  E ) ) ) : ( A  \ 
( `' f " ran  E ) ) -onto-> ( om  \  ran  E
) )
79 foco 5663 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( ( `' E  o.  `' ( S  |`  ran  E ) )  o.  ( f  |`  ( A  \  ( `' f
" ran  E )
) ) ) : ( A  \  ( `' f " ran  E ) ) -onto-> om )
81 fowdom 7539 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( A  \  ( `' f " ran  E ) ) )
83 difexg 4351 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  \  ( `' f
" ran  E )
)  e.  _V )
84 isfin3-2 8247 . . . . . . . . . . 11  |-  ( ( A  \  ( `' f " ran  E
) )  e.  _V  ->  ( ( A  \ 
( `' f " ran  E ) )  e. FinIII  <->  -.  om  ~<_*  ( A  \  ( `' f " ran  E ) ) ) )
8512, 83, 843syl 19 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( ( A  \  ( `' f " ran  E ) )  e. FinIII  <->  -.  om  ~<_*  ( A  \  ( `' f " ran  E ) ) ) )
8685con2bid 320 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( om  ~<_*  ( A  \  ( `' f " ran  E ) )  <->  -.  ( A  \  ( `' f
" ran  E )
)  e. FinIII ) )
8782, 86mpbid 202 . . . . . . . 8  |-  ( f : A -onto-> om  ->  -.  ( A  \  ( `' f " ran  E ) )  e. FinIII )
88 eleq1 2496 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( y  e. FinIII  <->  ( `' f " ran  E )  e. FinIII ) )
89 difeq2 3459 . . . . . . . . . . . . 13  |-  ( y  =  ( `' f
" ran  E )  ->  ( A  \  y
)  =  ( A 
\  ( `' f
" ran  E )
) )
9089eleq1d 2502 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( ( A  \ 
y )  e. FinIII  <->  ( A  \  ( `' f " ran  E ) )  e. FinIII ) )
9188, 90orbi12d 691 . . . . . . . . . . 11  |-  ( y  =  ( `' f
" ran  E )  ->  ( ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII )  <->  ( ( `' f " ran  E )  e. FinIII  \/  ( A  \  ( `' f " ran  E ) )  e. FinIII ) ) )
9291notbid 286 . . . . . . . . . 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 477 . . . . . . . . . 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 253 . . . . . . . . 9  |-  ( y  =  ( `' f
" ran  E )  ->  ( -.  ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII )  <->  ( -.  ( `' f " ran  E )  e. FinIII  /\  -.  ( A  \  ( `' f
" ran  E )
)  e. FinIII ) ) )
9594rspcev 3052 . . . . . . . 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 1182 . . . . . . 7  |-  ( f : A -onto-> om  ->  E. y  e.  ~P  A  -.  ( y  e. FinIII  \/  ( A  \  y )  e. FinIII ) )
97 rexnal 2716 . . . . . . 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 189 . . . . . 6  |-  ( f : A -onto-> om  ->  -. 
A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII ) )
9998exlimiv 1644 . . . . 5  |-  ( E. f  f : A -onto-> om  ->  -.  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII ) )
1004, 99sylbi 188 . . . 4  |-  ( om  ~<_*  A  ->  -.  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \ 
y )  e. FinIII ) )
101100con2i 114 . . 3  |-  ( A. y  e.  ~P  A
( y  e. FinIII  \/  ( A  \  y )  e. FinIII )  ->  -.  om  ~<_*  A )
102 isfin3-2 8247 . . 3  |-  ( A  e.  V  ->  ( A  e. FinIII 
<->  -.  om  ~<_*  A ) )
103101, 102syl5ibr 213 . 2  |-  ( A  e.  V  ->  ( A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
)  e. FinIII )  ->  A  e. FinIII ) )
104103imp 419 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 177    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   class class class wbr 4212    e. cmpt 4266   Oncon0 4581   suc csuc 4583   omcom 4845   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881    o. ccom 4882   Fun wfun 5448   -->wf 5450   -1-1->wf1 5451   -onto->wfo 5452   -1-1-onto->wf1o 5453  (class class class)co 6081   2oc2o 6718    .o comu 6722    ~<_* cwdom 7525  FinIIIcfin3 8161
This theorem is referenced by:  fin1a2lem8  8287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-seqom 6705  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-wdom 7527  df-card 7826  df-fin4 8167  df-fin3 8168
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