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Theorem phplem4on 6712
Description: Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
Assertion
Ref Expression
phplem4on  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )

Proof of Theorem phplem4on
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 bren 6593 . . . . 5  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
21biimpi 119 . . . 4  |-  ( suc 
A  ~~  suc  B  ->  E. f  f : suc  A -1-1-onto-> suc  B )
32adantl 273 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  suc  A  ~~  suc  B )  ->  E. f 
f : suc  A -1-1-onto-> suc  B )
4 f1of1 5320 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
54adantl 273 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  f : suc  A -1-1-> suc  B )
6 peano2 4467 . . . . . . . . 9  |-  ( B  e.  om  ->  suc  B  e.  om )
7 nnon 4481 . . . . . . . . 9  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
86, 7syl 14 . . . . . . . 8  |-  ( B  e.  om  ->  suc  B  e.  On )
98ad3antlr 482 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  suc  B  e.  On )
10 sssucid 4295 . . . . . . . 8  |-  A  C_  suc  A
1110a1i 9 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  C_  suc  A )
12 simplll 505 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  e.  On )
13 f1imaen2g 6639 . . . . . . 7  |-  ( ( ( f : suc  A
-1-1-> suc  B  /\  suc  B  e.  On )  /\  ( A  C_  suc  A  /\  A  e.  On ) )  ->  (
f " A ) 
~~  A )
145, 9, 11, 12, 13syl22anc 1198 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " A )  ~~  A
)
1514ensymd 6629 . . . . 5  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  ( f " A
) )
16 eloni 4255 . . . . . . . . 9  |-  ( A  e.  On  ->  Ord  A )
17 orddif 4420 . . . . . . . . 9  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1816, 17syl 14 . . . . . . . 8  |-  ( A  e.  On  ->  A  =  ( suc  A  \  { A } ) )
1918imaeq2d 4837 . . . . . . 7  |-  ( A  e.  On  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
2019ad3antrrr 481 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " A )  =  ( f " ( suc 
A  \  { A } ) ) )
21 f1ofn 5322 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
2221adantl 273 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  f  Fn  suc  A )
23 sucidg 4296 . . . . . . . . . 10  |-  ( A  e.  On  ->  A  e.  suc  A )
2412, 23syl 14 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  e.  suc  A )
25 fnsnfv 5432 . . . . . . . . 9  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2622, 24, 25syl2anc 406 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2726difeq2d 3158 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( (
f " suc  A
)  \  { (
f `  A ) } )  =  ( ( f " suc  A )  \  ( f
" { A }
) ) )
28 imadmrn 4847 . . . . . . . . . . 11  |-  ( f
" dom  f )  =  ran  f
2928eqcomi 2117 . . . . . . . . . 10  |-  ran  f  =  ( f " dom  f )
30 f1ofo 5328 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
31 forn 5304 . . . . . . . . . . 11  |-  ( f : suc  A -onto-> suc  B  ->  ran  f  =  suc  B )
3230, 31syl 14 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ran  f  =  suc  B )
33 f1odm 5325 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
3433imaeq2d 4837 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
3529, 32, 343eqtr3a 2169 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
3635difeq1d 3157 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
3736adantl 273 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( suc  B 
\  { ( f `
 A ) } )  =  ( ( f " suc  A
)  \  { (
f `  A ) } ) )
38 dff1o3 5327 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3938simprbi 271 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
40 imadif 5159 . . . . . . . . 9  |-  ( Fun  `' f  ->  ( f
" ( suc  A  \  { A } ) )  =  ( ( f " suc  A
)  \  ( f " { A } ) ) )
4139, 40syl 14 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
4241adantl 273 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " ( suc  A  \  { A } ) )  =  ( ( f " suc  A
)  \  ( f " { A } ) ) )
4327, 37, 423eqtr4rd 2156 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " ( suc  A  \  { A } ) )  =  ( suc 
B  \  { (
f `  A ) } ) )
4420, 43eqtrd 2145 . . . . 5  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " A )  =  ( suc  B  \  {
( f `  A
) } ) )
4515, 44breqtrd 3917 . . . 4  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  ( suc  B  \  {
( f `  A
) } ) )
46 simpllr 506 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  B  e.  om )
47 fnfvelrn 5504 . . . . . . . 8  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
4822, 24, 47syl2anc 406 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f `  A )  e.  ran  f )
4931eleq2d 2182 . . . . . . . . 9  |-  ( f : suc  A -onto-> suc  B  ->  ( ( f `
 A )  e. 
ran  f  <->  ( f `  A )  e.  suc  B ) )
5030, 49syl 14 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f `  A )  e.  ran  f 
<->  ( f `  A
)  e.  suc  B
) )
5150adantl 273 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( (
f `  A )  e.  ran  f  <->  ( f `  A )  e.  suc  B ) )
5248, 51mpbid 146 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f `  A )  e.  suc  B )
53 phplem3g 6701 . . . . . 6  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
5446, 52, 53syl2anc 406 . . . . 5  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
5554ensymd 6629 . . . 4  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( suc  B 
\  { ( f `
 A ) } )  ~~  B )
56 entr 6630 . . . 4  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
5745, 55, 56syl2anc 406 . . 3  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
583, 57exlimddv 1850 . 2  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  suc  A  ~~  suc  B )  ->  A  ~~  B )
5958ex 114 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1312   E.wex 1449    e. wcel 1461    \ cdif 3032    C_ wss 3035   {csn 3491   class class class wbr 3893   Ord word 4242   Oncon0 4243   suc csuc 4245   omcom 4462   `'ccnv 4496   dom cdm 4497   ran crn 4498   "cima 4500   Fun wfun 5073    Fn wfn 5074   -1-1->wf1 5076   -onto->wfo 5077   -1-1-onto->wf1o 5078   ` cfv 5079    ~~ cen 6584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-er 6381  df-en 6587
This theorem is referenced by: (None)
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