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Theorem r1om 8017
Description: The set of hereditarily finite sets is countable. See ackbij2 8016 for an explicit bijection that works without Infinity. See also r1omALT 8545. (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 7491 . . . 4  |-  om  e.  _V
2 limom 4774 . . . 4  |-  Lim  om
3 r1lim 7591 . . . 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 7586 . . . 4  |-  R1  Fn  On
6 fnfun 5446 . . . 4  |-  ( R1  Fn  On  ->  Fun  R1 )
7 funiunfv 5895 . . . 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 2386 . 2  |-  ( R1
`  om )  =  U. ( R1 " om )
10 iuneq1 4020 . . . . . . 7  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ f  e.  a  ( { f }  X.  ~P f ) )
11 sneq 3740 . . . . . . . . 9  |-  ( f  =  b  ->  { f }  =  { b } )
12 pweq 3717 . . . . . . . . 9  |-  ( f  =  b  ->  ~P f  =  ~P b
)
1311, 12xpeq12d 4817 . . . . . . . 8  |-  ( f  =  b  ->  ( { f }  X.  ~P f )  =  ( { b }  X.  ~P b ) )
1413cbviunv 4043 . . . . . . 7  |-  U_ f  e.  a  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b )
1510, 14syl6eq 2414 . . . . . 6  |-  ( e  =  a  ->  U_ f  e.  e  ( {
f }  X.  ~P f )  =  U_ b  e.  a  ( { b }  X.  ~P b ) )
1615fveq2d 5636 . . . . 5  |-  ( e  =  a  ->  ( card `  U_ f  e.  e  ( { f }  X.  ~P f
) )  =  (
card `  U_ b  e.  a  ( { b }  X.  ~P b
) ) )
1716cbvmptv 4213 . . . 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 4982 . . . . . . . 8  |-  ( c  =  a  ->  dom  c  =  dom  a )
1918pweqd 3719 . . . . . . 7  |-  ( c  =  a  ->  ~P dom  c  =  ~P dom  a )
20 imaeq1 5110 . . . . . . . 8  |-  ( c  =  a  ->  (
c " d )  =  ( a "
d ) )
2120fveq2d 5636 . . . . . . 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 4200 . . . . . 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 5111 . . . . . . . 8  |-  ( d  =  b  ->  (
a " d )  =  ( a "
b ) )
2423fveq2d 5636 . . . . . . 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 4213 . . . . . 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 2414 . . . . 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 4213 . . . 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 2366 . . . 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 8016 . . 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 5646 . . . . 5  |-  ( R1
`  om )  e.  _V
319, 30eqeltrri 2437 . . . 4  |-  U. ( R1 " om )  e. 
_V
3231f1oen 7025 . . 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 4144 1  |-  ( R1
`  om )  ~~  om
Colors of variables: wff set class
Syntax hints:    = wceq 1647    e. wcel 1715   _Vcvv 2873    i^i cin 3237   (/)c0 3543   ~Pcpw 3714   {csn 3729   U.cuni 3929   U_ciun 4007   class class class wbr 4125    e. cmpt 4179   Oncon0 4495   Lim wlim 4496   omcom 4759    X. cxp 4790   dom cdm 4792   "cima 4795   Fun wfun 5352    Fn wfn 5353   -1-1-onto->wf1o 5357   ` cfv 5358   reccrdg 6564    ~~ cen 7003   Fincfn 7006   R1cr1 7581   cardccrd 7715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-r1 7583  df-rank 7584  df-card 7719  df-cda 7941
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