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Theorem r1om 8088
Description: The set of hereditarily finite sets is countable. See ackbij2 8087 for an explicit bijection that works without Infinity. See also r1omALT 8615. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
r1om  |-  ( R1
`  om )  ~~  om

Proof of Theorem r1om
Dummy variables  a 
b  c  d  e  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 7562 . . . 4  |-  om  e.  _V
2 limom 4827 . . . 4  |-  Lim  om
3 r1lim 7662 . . . 4  |-  ( ( om  e.  _V  /\  Lim  om )  ->  ( R1 `  om )  = 
U_ a  e.  om  ( R1 `  a ) )
41, 2, 3mp2an 654 . . 3  |-  ( R1
`  om )  =  U_ a  e.  om  ( R1 `  a )
5 r1fnon 7657 . . . 4  |-  R1  Fn  On
6 fnfun 5509 . . . 4  |-  ( R1  Fn  On  ->  Fun  R1 )
7 funiunfv 5962 . . . 4  |-  ( Fun 
R1  ->  U_ a  e.  om  ( R1 `  a )  =  U. ( R1
" om ) )
85, 6, 7mp2b 10 . . 3  |-  U_ a  e.  om  ( R1 `  a )  =  U. ( R1 " om )
94, 8eqtri 2432 . 2  |-  ( R1
`  om )  =  U. ( R1 " om )
10 iuneq1 4074 . . . . . . 7  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ f  e.  a  ( { f }  X.  ~P f ) )
11 sneq 3793 . . . . . . . . 9  |-  ( f  =  b  ->  { f }  =  { b } )
12 pweq 3770 . . . . . . . . 9  |-  ( f  =  b  ->  ~P f  =  ~P b
)
1311, 12xpeq12d 4870 . . . . . . . 8  |-  ( f  =  b  ->  ( { f }  X.  ~P f )  =  ( { b }  X.  ~P b ) )
1413cbviunv 4098 . . . . . . 7  |-  U_ f  e.  a  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b )
1510, 14syl6eq 2460 . . . . . 6  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b ) )
1615fveq2d 5699 . . . . 5  |-  ( e  =  a  ->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f
) )  =  (
card `  U_ b  e.  a  ( { b }  X.  ~P b
) ) )
1716cbvmptv 4268 . . . 4  |-  ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f ) ) )  =  ( a  e.  ( ~P
om  i^i  Fin )  |->  ( card `  U_ b  e.  a  ( {
b }  X.  ~P b ) ) )
18 dmeq 5037 . . . . . . . 8  |-  ( c  =  a  ->  dom  c  =  dom  a )
1918pweqd 3772 . . . . . . 7  |-  ( c  =  a  ->  ~P dom  c  =  ~P dom  a )
20 imaeq1 5165 . . . . . . . 8  |-  ( c  =  a  ->  (
c " d )  =  ( a "
d ) )
2120fveq2d 5699 . . . . . . 7  |-  ( c  =  a  ->  (
( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( c "
d ) )  =  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( a "
d ) ) )
2219, 21mpteq12dv 4255 . . . . . 6  |-  ( c  =  a  ->  (
d  e.  ~P dom  c  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f ) ) ) `  (
c " d ) ) )  =  ( d  e.  ~P dom  a  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f ) ) ) `  (
a " d ) ) ) )
23 imaeq2 5166 . . . . . . . 8  |-  ( d  =  b  ->  (
a " d )  =  ( a "
b ) )
2423fveq2d 5699 . . . . . . 7  |-  ( d  =  b  ->  (
( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( a "
d ) )  =  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( a "
b ) ) )
2524cbvmptv 4268 . . . . . 6  |-  ( d  e.  ~P dom  a  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( a "
d ) ) )  =  ( b  e. 
~P dom  a  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( a "
b ) ) )
2622, 25syl6eq 2460 . . . . 5  |-  ( c  =  a  ->  (
d  e.  ~P dom  c  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f ) ) ) `  (
c " d ) ) )  =  ( b  e.  ~P dom  a  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f ) ) ) `  (
a " b ) ) ) )
2726cbvmptv 4268 . . . 4  |-  ( c  e.  _V  |->  ( d  e.  ~P dom  c  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( c "
d ) ) ) )  =  ( a  e.  _V  |->  ( b  e.  ~P dom  a  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( a "
b ) ) ) )
28 eqid 2412 . . . 4  |-  U. ( rec ( ( c  e. 
_V  |->  ( d  e. 
~P dom  c  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( c "
d ) ) ) ) ,  (/) ) " om )  =  U. ( rec ( ( c  e.  _V  |->  ( d  e.  ~P dom  c  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( c "
d ) ) ) ) ,  (/) ) " om )
2917, 27, 28ackbij2 8087 . . 3  |-  U. ( rec ( ( c  e. 
_V  |->  ( d  e. 
~P dom  c  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( c "
d ) ) ) ) ,  (/) ) " om ) : U. ( R1 " om ) -1-1-onto-> om
30 fvex 5709 . . . . 5  |-  ( R1
`  om )  e.  _V
319, 30eqeltrri 2483 . . . 4  |-  U. ( R1 " om )  e. 
_V
3231f1oen 7095 . . 3  |-  ( U. ( rec ( ( c  e.  _V  |->  ( d  e.  ~P dom  c  |->  ( ( e  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ f  e.  e  ( {
f }  X.  ~P f ) ) ) `
 ( c "
d ) ) ) ) ,  (/) ) " om ) : U. ( R1 " om ) -1-1-onto-> om  ->  U. ( R1 " om )  ~~  om )
3329, 32ax-mp 8 . 2  |-  U. ( R1 " om )  ~~  om
349, 33eqbrtri 4199 1  |-  ( R1
`  om )  ~~  om
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2924    i^i cin 3287   (/)c0 3596   ~Pcpw 3767   {csn 3782   U.cuni 3983   U_ciun 4061   class class class wbr 4180    e. cmpt 4234   Oncon0 4549   Lim wlim 4550   omcom 4812    X. cxp 4843   dom cdm 4845   "cima 4848   Fun wfun 5415    Fn wfn 5416   -1-1-onto->wf1o 5420   ` cfv 5421   reccrdg 6634    ~~ cen 7073   Fincfn 7076   R1cr1 7652   cardccrd 7786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-r1 7654  df-rank 7655  df-card 7790  df-cda 8012
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