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Theorem fin1a2lem7 8782
Description: Lemma for fin1a2 8791. 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 6697 . . . . . 6  |-  (/)  e.  om
2 ne0i 3791 . . . . . 6  |-  ( (/)  e.  om  ->  om  =/=  (/) )
3 brwdomn0 7991 . . . . . 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 5355 . . . . . . . . . 10  |-  ( `' f " ran  E
)  C_  dom  f
6 fof 5793 . . . . . . . . . . 11  |-  ( f : A -onto-> om  ->  f : A --> om )
7 fdm 5733 . . . . . . . . . . 11  |-  ( f : A --> om  ->  dom  f  =  A )
86, 7syl 16 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  dom  f  =  A )
95, 8syl5sseq 3552 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  ( `' f " ran  E )  C_  A )
10 vex 3116 . . . . . . . . . . 11  |-  f  e. 
_V
11 dmfex 6739 . . . . . . . . . . 11  |-  ( ( f  e.  _V  /\  f : A --> om )  ->  A  e.  _V )
1210, 6, 11sylancr 663 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  A  e.  _V )
13 elpw2g 4610 . . . . . . . . . 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 8779 . . . . . . . . . . . . 13  |-  E : om
-1-1-> om
18 f1cnv 5837 . . . . . . . . . . . . 13  |-  ( E : om -1-1-> om  ->  `' E : ran  E -1-1-onto-> om )
19 f1ofo 5821 . . . . . . . . . . . . 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 5794 . . . . . . . . . . . 12  |-  ( `' E : ran  E -onto-> om  ->  Fun  `' E
)
2220, 21ax-mp 5 . . . . . . . . . . 11  |-  Fun  `' E
2310resex 5315 . . . . . . . . . . 11  |-  ( f  |`  ( `' f " ran  E ) )  e. 
_V
24 cofunexg 6745 . . . . . . . . . . 11  |-  ( ( Fun  `' E  /\  ( f  |`  ( `' f " ran  E ) )  e.  _V )  ->  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e. 
_V )
2522, 23, 24mp2an 672 . . . . . . . . . 10  |-  ( `' E  o.  ( f  |`  ( `' f " ran  E ) ) )  e.  _V
26 fofun 5794 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  Fun  f )
27 fores 5802 . . . . . . . . . . . . 13  |-  ( ( Fun  f  /\  ( `' f " ran  E )  C_  dom  f )  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E ) -onto-> ( f "
( `' f " ran  E ) ) )
2826, 5, 27sylancl 662 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( `' f " ran  E ) ) : ( `' f " ran  E
) -onto-> ( f "
( `' f " ran  E ) ) )
29 f1f 5779 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  E : om --> om )
30 frn 5735 . . . . . . . . . . . . . . 15  |-  ( E : om --> om  ->  ran 
E  C_  om )
3117, 29, 30mp2b 10 . . . . . . . . . . . . . 14  |-  ran  E  C_ 
om
32 foimacnv 5831 . . . . . . . . . . . . . 14  |-  ( ( f : A -onto-> om  /\ 
ran  E  C_  om )  ->  ( f " ( `' f " ran  E ) )  =  ran  E )
3331, 32mpan2 671 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ran  E
) )  =  ran  E )
34 foeq3 5791 . . . . . . . . . . . . 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 5803 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( `' E  o.  (
f  |`  ( `' f
" ran  E )
) ) : ( `' f " ran  E ) -onto-> om )
39 fowdom 7993 . . . . . . . . . 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 663 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( `' f " ran  E ) )
41 cnvexg 6727 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  `' f  e.  _V )
42 imaexg 6718 . . . . . . . . . . . 12  |-  ( `' f  e.  _V  ->  ( `' f " ran  E )  e.  _V )
4310, 41, 42mp2b 10 . . . . . . . . . . 11  |-  ( `' f " ran  E
)  e.  _V
44 isfin3-2 8743 . . . . . . . . . . 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 8781 . . . . . . . . . . . . . 14  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
50 f1ocnv 5826 . . . . . . . . . . . . . 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 5821 . . . . . . . . . . . . . 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 5803 . . . . . . . . . . . . 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 672 . . . . . . . . . . . 12  |-  ( `' E  o.  `' ( S  |`  ran  E ) ) : ( om 
\  ran  E ) -onto-> om
55 fofun 5794 . . . . . . . . . . . 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 5315 . . . . . . . . . . 11  |-  ( f  |`  ( A  \  ( `' f " ran  E ) ) )  e. 
_V
58 cofunexg 6745 . . . . . . . . . . 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 672 . . . . . . . . . 10  |-  ( ( `' E  o.  `' ( S  |`  ran  E
) )  o.  (
f  |`  ( A  \ 
( `' f " ran  E ) ) ) )  e.  _V
60 difss 3631 . . . . . . . . . . . . . 14  |-  ( A 
\  ( `' f
" ran  E )
)  C_  A
6160, 8syl5sseqr 3553 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( A  \  ( `' f " ran  E
) )  C_  dom  f )
62 fores 5802 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 12  |-  ( f : A -onto-> om  ->  ( f  |`  ( A  \  ( `' f " ran  E ) ) ) : ( A  \ 
( `' f " ran  E ) ) -onto-> ( f " ( A 
\  ( `' f
" ran  E )
) ) )
64 funcnvcnv 5644 . . . . . . . . . . . . . . . 16  |-  ( Fun  f  ->  Fun  `' `' f )
65 imadif 5661 . . . . . . . . . . . . . . . 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 5335 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) ) )
68 difss 3631 . . . . . . . . . . . . . . 15  |-  ( om 
\  ran  E )  C_ 
om
69 foimacnv 5831 . . . . . . . . . . . . . . 15  |-  ( ( f : A -onto-> om  /\  ( om  \  ran  E )  C_  om )  ->  ( f " ( `' f " ( om  \  ran  E ) ) )  =  ( om  \  ran  E
) )
7068, 69mpan2 671 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( `' f " ( om 
\  ran  E )
) )  =  ( om  \  ran  E
) )
71 fimacnv 6011 . . . . . . . . . . . . . . . . 17  |-  ( f : A --> om  ->  ( `' f " om )  =  A )
726, 71syl 16 . . . . . . . . . . . . . . . 16  |-  ( f : A -onto-> om  ->  ( `' f " om )  =  A )
7372difeq1d 3621 . . . . . . . . . . . . . . 15  |-  ( f : A -onto-> om  ->  ( ( `' f " om )  \  ( `' f " ran  E ) )  =  ( A  \  ( `' f " ran  E
) ) )
7473imaeq2d 5335 . . . . . . . . . . . . . 14  |-  ( f : A -onto-> om  ->  ( f " ( ( `' f " om )  \  ( `' f
" ran  E )
) )  =  ( f " ( A 
\  ( `' f
" ran  E )
) ) )
7567, 70, 743eqtr3rd 2517 . . . . . . . . . . . . 13  |-  ( f : A -onto-> om  ->  ( f " ( A 
\  ( `' f
" ran  E )
) )  =  ( om  \  ran  E
) )
76 foeq3 5791 . . . . . . . . . . . . 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 5803 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( f : A -onto-> om  ->  ( ( `' E  o.  `' ( S  |`  ran  E ) )  o.  ( f  |`  ( A  \  ( `' f
" ran  E )
) ) ) : ( A  \  ( `' f " ran  E ) ) -onto-> om )
81 fowdom 7993 . . . . . . . . . 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 663 . . . . . . . . 9  |-  ( f : A -onto-> om  ->  om  ~<_*  ( A  \  ( `' f " ran  E ) ) )
83 difexg 4595 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  \  ( `' f
" ran  E )
)  e.  _V )
84 isfin3-2 8743 . . . . . . . . . . 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 2539 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( y  e. FinIII  <->  ( `' f " ran  E )  e. FinIII ) )
89 difeq2 3616 . . . . . . . . . . . . 13  |-  ( y  =  ( `' f
" ran  E )  ->  ( A  \  y
)  =  ( A 
\  ( `' f
" ran  E )
) )
9089eleq1d 2536 . . . . . . . . . . . 12  |-  ( y  =  ( `' f
" ran  E )  ->  ( ( A  \ 
y )  e. FinIII  <->  ( A  \  ( `' f " ran  E ) )  e. FinIII ) )
9188, 90orbi12d 709 . . . . . . . . . . 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 490 . . . . . . . . . 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 3214 . . . . . . . 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 1226 . . . . . . 7  |-  ( f : A -onto-> om  ->  E. y  e.  ~P  A  -.  ( y  e. FinIII  \/  ( A  \  y )  e. FinIII ) )
97 rexnal 2912 . . . . . . 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 1698 . . . . 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 8743 . . 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   Oncon0 4878   suc csuc 4880   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    o. ccom 5003   Fun wfun 5580   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585  (class class class)co 6282   omcom 6678   2oc2o 7121    .o comu 7125    ~<_* cwdom 7979  FinIIIcfin3 8657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-seqom 7110  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-wdom 7981  df-card 8316  df-fin4 8663  df-fin3 8664
This theorem is referenced by:  fin1a2lem8  8783
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