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Theorem r1om 8613
Description: The set of hereditarily finite sets is countable. See ackbij2 8612 for an explicit bijection that works without Infinity. See also r1omALT 9143. (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 8049 . . . 4  |-  om  e.  _V
2 limom 6686 . . . 4  |-  Lim  om
3 r1lim 8179 . . . 4  |-  ( ( om  e.  _V  /\  Lim  om )  ->  ( R1 `  om )  = 
U_ a  e.  om  ( R1 `  a ) )
41, 2, 3mp2an 672 . . 3  |-  ( R1
`  om )  =  U_ a  e.  om  ( R1 `  a )
5 r1fnon 8174 . . . 4  |-  R1  Fn  On
6 fnfun 5669 . . . 4  |-  ( R1  Fn  On  ->  Fun  R1 )
7 funiunfv 6139 . . . 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 2489 . 2  |-  ( R1
`  om )  =  U. ( R1 " om )
10 iuneq1 4332 . . . . . . 7  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ f  e.  a  ( { f }  X.  ~P f ) )
11 sneq 4030 . . . . . . . . 9  |-  ( f  =  b  ->  { f }  =  { b } )
12 pweq 4006 . . . . . . . . 9  |-  ( f  =  b  ->  ~P f  =  ~P b
)
1311, 12xpeq12d 5017 . . . . . . . 8  |-  ( f  =  b  ->  ( { f }  X.  ~P f )  =  ( { b }  X.  ~P b ) )
1413cbviunv 4357 . . . . . . 7  |-  U_ f  e.  a  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b )
1510, 14syl6eq 2517 . . . . . 6  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b ) )
1615fveq2d 5861 . . . . 5  |-  ( e  =  a  ->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f
) )  =  (
card `  U_ b  e.  a  ( { b }  X.  ~P b
) ) )
1716cbvmptv 4531 . . . 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 5194 . . . . . . . 8  |-  ( c  =  a  ->  dom  c  =  dom  a )
1918pweqd 4008 . . . . . . 7  |-  ( c  =  a  ->  ~P dom  c  =  ~P dom  a )
20 imaeq1 5323 . . . . . . . 8  |-  ( c  =  a  ->  (
c " d )  =  ( a "
d ) )
2120fveq2d 5861 . . . . . . 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 4518 . . . . . 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 5324 . . . . . . . 8  |-  ( d  =  b  ->  (
a " d )  =  ( a "
b ) )
2423fveq2d 5861 . . . . . . 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 4531 . . . . . 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 2517 . . . . 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 4531 . . . 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 2460 . . . 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 8612 . . 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 5867 . . . . 5  |-  ( R1
`  om )  e.  _V
319, 30eqeltrri 2545 . . . 4  |-  U. ( R1 " om )  e. 
_V
3231f1oen 7526 . . 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 5 . 2  |-  U. ( R1 " om )  ~~  om
349, 33eqbrtri 4459 1  |-  ( R1
`  om )  ~~  om
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   _Vcvv 3106    i^i cin 3468   (/)c0 3778   ~Pcpw 4003   {csn 4020   U.cuni 4238   U_ciun 4318   class class class wbr 4440    |-> cmpt 4498   Oncon0 4871   Lim wlim 4872    X. cxp 4990   dom cdm 4992   "cima 4995   Fun wfun 5573    Fn wfn 5574   -1-1-onto->wf1o 5578   ` cfv 5579   omcom 6671   reccrdg 7065    ~~ cen 7503   Fincfn 7506   R1cr1 8169   cardccrd 8305
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-r1 8171  df-rank 8172  df-card 8309  df-cda 8537
This theorem is referenced by: (None)
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