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Theorem ssnnf1octb 37470
Description: There exists a bijection between a subset of  NN and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Assertion
Ref Expression
ssnnf1octb  |-  ( ( A  ~<_  om  /\  A  =/=  (/) )  ->  E. f
( dom  f  C_  NN  /\  f : dom  f
-1-1-onto-> A ) )
Distinct variable group:    A, f

Proof of Theorem ssnnf1octb
Dummy variables  g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnfoctb 37383 . 2  |-  ( ( A  ~<_  om  /\  A  =/=  (/) )  ->  E. g 
g : NN -onto-> A
)
2 fofn 5795 . . . . . 6  |-  ( g : NN -onto-> A  -> 
g  Fn  NN )
3 nnex 10615 . . . . . . 7  |-  NN  e.  _V
43a1i 11 . . . . . 6  |-  ( g : NN -onto-> A  ->  NN  e.  _V )
5 ltwenn 12176 . . . . . . 7  |-  <  We  NN
65a1i 11 . . . . . 6  |-  ( g : NN -onto-> A  ->  <  We  NN )
72, 4, 6wessf1orn 37460 . . . . 5  |-  ( g : NN -onto-> A  ->  E. x  e.  ~P  NN ( g  |`  x
) : x -1-1-onto-> ran  g
)
8 f1odm 5818 . . . . . . . . . . 11  |-  ( ( g  |`  x ) : x -1-1-onto-> ran  g  ->  dom  ( g  |`  x
)  =  x )
98adantl 468 . . . . . . . . . 10  |-  ( ( x  e.  ~P NN  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  dom  ( g  |`  x )  =  x )
10 elpwi 3960 . . . . . . . . . . 11  |-  ( x  e.  ~P NN  ->  x 
C_  NN )
1110adantr 467 . . . . . . . . . 10  |-  ( ( x  e.  ~P NN  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  x  C_  NN )
129, 11eqsstrd 3466 . . . . . . . . 9  |-  ( ( x  e.  ~P NN  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  dom  ( g  |`  x )  C_  NN )
13123adant1 1026 . . . . . . . 8  |-  ( ( g : NN -onto-> A  /\  x  e.  ~P NN  /\  ( g  |`  x ) : x -1-1-onto-> ran  g )  ->  dom  ( g  |`  x
)  C_  NN )
14 simpr 463 . . . . . . . . . 10  |-  ( ( g : NN -onto-> A  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  ( g  |`  x ) : x -1-1-onto-> ran  g )
15 eqidd 2452 . . . . . . . . . . 11  |-  ( ( g : NN -onto-> A  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  ( g  |`  x )  =  ( g  |`  x )
)
168eqcomd 2457 . . . . . . . . . . . 12  |-  ( ( g  |`  x ) : x -1-1-onto-> ran  g  ->  x  =  dom  ( g  |`  x ) )
1716adantl 468 . . . . . . . . . . 11  |-  ( ( g : NN -onto-> A  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  x  =  dom  ( g  |`  x
) )
18 forn 5796 . . . . . . . . . . . 12  |-  ( g : NN -onto-> A  ->  ran  g  =  A
)
1918adantr 467 . . . . . . . . . . 11  |-  ( ( g : NN -onto-> A  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  ran  g  =  A )
2015, 17, 19f1oeq123d 5811 . . . . . . . . . 10  |-  ( ( g : NN -onto-> A  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  ( (
g  |`  x ) : x -1-1-onto-> ran  g  <->  ( g  |`  x ) : dom  ( g  |`  x
)
-1-1-onto-> A ) )
2114, 20mpbid 214 . . . . . . . . 9  |-  ( ( g : NN -onto-> A  /\  ( g  |`  x
) : x -1-1-onto-> ran  g
)  ->  ( g  |`  x ) : dom  ( g  |`  x
)
-1-1-onto-> A )
22213adant2 1027 . . . . . . . 8  |-  ( ( g : NN -onto-> A  /\  x  e.  ~P NN  /\  ( g  |`  x ) : x -1-1-onto-> ran  g )  ->  (
g  |`  x ) : dom  ( g  |`  x ) -1-1-onto-> A )
23 vex 3048 . . . . . . . . . 10  |-  g  e. 
_V
2423resex 5148 . . . . . . . . 9  |-  ( g  |`  x )  e.  _V
25 dmeq 5035 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  x )  ->  dom  f  =  dom  ( g  |`  x ) )
2625sseq1d 3459 . . . . . . . . . 10  |-  ( f  =  ( g  |`  x )  ->  ( dom  f  C_  NN  <->  dom  ( g  |`  x )  C_  NN ) )
27 id 22 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  x )  ->  f  =  ( g  |`  x ) )
28 eqidd 2452 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  x )  ->  A  =  A )
2927, 25, 28f1oeq123d 5811 . . . . . . . . . 10  |-  ( f  =  ( g  |`  x )  ->  (
f : dom  f -1-1-onto-> A  <->  ( g  |`  x ) : dom  ( g  |`  x ) -1-1-onto-> A ) )
3026, 29anbi12d 717 . . . . . . . . 9  |-  ( f  =  ( g  |`  x )  ->  (
( dom  f  C_  NN  /\  f : dom  f
-1-1-onto-> A )  <->  ( dom  ( g  |`  x
)  C_  NN  /\  (
g  |`  x ) : dom  ( g  |`  x ) -1-1-onto-> A ) ) )
3124, 30spcev 3141 . . . . . . . 8  |-  ( ( dom  ( g  |`  x )  C_  NN  /\  ( g  |`  x
) : dom  (
g  |`  x ) -1-1-onto-> A )  ->  E. f ( dom  f  C_  NN  /\  f : dom  f -1-1-onto-> A ) )
3213, 22, 31syl2anc 667 . . . . . . 7  |-  ( ( g : NN -onto-> A  /\  x  e.  ~P NN  /\  ( g  |`  x ) : x -1-1-onto-> ran  g )  ->  E. f
( dom  f  C_  NN  /\  f : dom  f
-1-1-onto-> A ) )
33323exp 1207 . . . . . 6  |-  ( g : NN -onto-> A  -> 
( x  e.  ~P NN  ->  ( ( g  |`  x ) : x -1-1-onto-> ran  g  ->  E. f
( dom  f  C_  NN  /\  f : dom  f
-1-1-onto-> A ) ) ) )
3433rexlimdv 2877 . . . . 5  |-  ( g : NN -onto-> A  -> 
( E. x  e. 
~P  NN ( g  |`  x ) : x -1-1-onto-> ran  g  ->  E. f
( dom  f  C_  NN  /\  f : dom  f
-1-1-onto-> A ) ) )
357, 34mpd 15 . . . 4  |-  ( g : NN -onto-> A  ->  E. f ( dom  f  C_  NN  /\  f : dom  f -1-1-onto-> A ) )
3635a1i 11 . . 3  |-  ( ( A  ~<_  om  /\  A  =/=  (/) )  ->  ( g : NN -onto-> A  ->  E. f ( dom  f  C_  NN  /\  f : dom  f -1-1-onto-> A ) ) )
3736exlimdv 1779 . 2  |-  ( ( A  ~<_  om  /\  A  =/=  (/) )  ->  ( E. g  g : NN -onto-> A  ->  E. f ( dom  f  C_  NN  /\  f : dom  f -1-1-onto-> A ) ) )
381, 37mpd 15 1  |-  ( ( A  ~<_  om  /\  A  =/=  (/) )  ->  E. f
( dom  f  C_  NN  /\  f : dom  f
-1-1-onto-> A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   class class class wbr 4402    We wwe 4792   dom cdm 4834   ran crn 4835    |` cres 4836   -onto->wfo 5580   -1-1-onto->wf1o 5581   omcom 6692    ~<_ cdom 7567    < clt 9675   NNcn 10609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160
This theorem is referenced by:  isomennd  38352
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