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Theorem r1om 7870
Description: The set of hereditarily finite sets is countable. See ackbij2 7869 for an explicit bijection that works without Infinity. See also r1omALT 8398. (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 7344 . . . 4  |-  om  e.  _V
2 limom 4671 . . . 4  |-  Lim  om
3 r1lim 7444 . . . 4  |-  ( ( om  e.  _V  /\  Lim  om )  ->  ( R1 `  om )  = 
U_ a  e.  om  ( R1 `  a ) )
41, 2, 3mp2an 653 . . 3  |-  ( R1
`  om )  =  U_ a  e.  om  ( R1 `  a )
5 r1fnon 7439 . . . 4  |-  R1  Fn  On
6 fnfun 5341 . . . 4  |-  ( R1  Fn  On  ->  Fun  R1 )
7 funiunfv 5774 . . . 4  |-  ( Fun 
R1  ->  U_ a  e.  om  ( R1 `  a )  =  U. ( R1
" om ) )
85, 6, 7mp2b 9 . . 3  |-  U_ a  e.  om  ( R1 `  a )  =  U. ( R1 " om )
94, 8eqtri 2303 . 2  |-  ( R1
`  om )  =  U. ( R1 " om )
10 iuneq1 3918 . . . . . . 7  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ f  e.  a  ( { f }  X.  ~P f ) )
11 sneq 3651 . . . . . . . . 9  |-  ( f  =  b  ->  { f }  =  { b } )
12 pweq 3628 . . . . . . . . 9  |-  ( f  =  b  ->  ~P f  =  ~P b
)
1311, 12xpeq12d 4714 . . . . . . . 8  |-  ( f  =  b  ->  ( { f }  X.  ~P f )  =  ( { b }  X.  ~P b ) )
1413cbviunv 3941 . . . . . . 7  |-  U_ f  e.  a  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b )
1510, 14syl6eq 2331 . . . . . 6  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b ) )
1615fveq2d 5529 . . . . 5  |-  ( e  =  a  ->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f
) )  =  (
card `  U_ b  e.  a  ( { b }  X.  ~P b
) ) )
1716cbvmptv 4111 . . . 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 4879 . . . . . . . 8  |-  ( c  =  a  ->  dom  c  =  dom  a )
1918pweqd 3630 . . . . . . 7  |-  ( c  =  a  ->  ~P dom  c  =  ~P dom  a )
20 imaeq1 5007 . . . . . . . 8  |-  ( c  =  a  ->  (
c " d )  =  ( a "
d ) )
2120fveq2d 5529 . . . . . . 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 4098 . . . . . 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 5008 . . . . . . . 8  |-  ( d  =  b  ->  (
a " d )  =  ( a "
b ) )
2423fveq2d 5529 . . . . . . 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 4111 . . . . . 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 2331 . . . . 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 4111 . . . 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 2283 . . . 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 7869 . . 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 5539 . . . . 5  |-  ( R1
`  om )  e.  _V
319, 30eqeltrri 2354 . . . 4  |-  U. ( R1 " om )  e. 
_V
3231f1oen 6882 . . 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 4042 1  |-  ( R1
`  om )  ~~  om
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   Oncon0 4392   Lim wlim 4393   omcom 4656    X. cxp 4687   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255   reccrdg 6422    ~~ cen 6860   Fincfn 6863   R1cr1 7434   cardccrd 7568
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  ax-inf2 7342
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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-r1 7436  df-rank 7437  df-card 7572  df-cda 7794
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