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Theorem phplem4on 6360
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 6259 . . . . 5  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
21biimpi 117 . . . 4  |-  ( suc 
A  ~~  suc  B  ->  E. f  f : suc  A -1-1-onto-> suc  B )
32adantl 266 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  suc  A  ~~  suc  B )  ->  E. f 
f : suc  A -1-1-onto-> suc  B )
4 f1of1 5153 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
54adantl 266 . . . . . . 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 4346 . . . . . . . . 9  |-  ( B  e.  om  ->  suc  B  e.  om )
7 nnon 4360 . . . . . . . . 9  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
86, 7syl 14 . . . . . . . 8  |-  ( B  e.  om  ->  suc  B  e.  On )
98ad3antlr 470 . . . . . . 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 4180 . . . . . . . 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 493 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  e.  On )
13 f1imaen2g 6304 . . . . . . 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 1147 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " A )  ~~  A
)
1514ensymd 6294 . . . . 5  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  ( f " A
) )
16 eloni 4140 . . . . . . . . 9  |-  ( A  e.  On  ->  Ord  A )
17 orddif 4299 . . . . . . . . 9  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1816, 17syl 14 . . . . . . . 8  |-  ( A  e.  On  ->  A  =  ( suc  A  \  { A } ) )
1918imaeq2d 4696 . . . . . . 7  |-  ( A  e.  On  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
2019ad3antrrr 469 . . . . . 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 5155 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
2221adantl 266 . . . . . . . . 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 4181 . . . . . . . . . 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 5260 . . . . . . . . 9  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2622, 24, 25syl2anc 397 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2726difeq2d 3090 . . . . . . 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 4706 . . . . . . . . . . 11  |-  ( f
" dom  f )  =  ran  f
2928eqcomi 2060 . . . . . . . . . 10  |-  ran  f  =  ( f " dom  f )
30 f1ofo 5161 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
31 forn 5137 . . . . . . . . . . 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 5158 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
3433imaeq2d 4696 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
3529, 32, 343eqtr3a 2112 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
3635difeq1d 3089 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
3736adantl 266 . . . . . . 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 5160 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3938simprbi 264 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
40 imadif 5007 . . . . . . . . 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 266 . . . . . . 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 2099 . . . . . 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 2088 . . . . 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 3816 . . . 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 494 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  B  e.  om )
47 fnfvelrn 5327 . . . . . . . 8  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
4822, 24, 47syl2anc 397 . . . . . . 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 2123 . . . . . . . . 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 266 . . . . . . 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 139 . . . . . 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 6350 . . . . . 6  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
5446, 52, 53syl2anc 397 . . . . 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 6294 . . . 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 6295 . . . 4  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
5745, 55, 56syl2anc 397 . . 3  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
583, 57exlimddv 1794 . 2  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  suc  A  ~~  suc  B )  ->  A  ~~  B )
5958ex 112 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409    \ cdif 2942    C_ wss 2945   {csn 3403   class class class wbr 3792   Ord word 4127   Oncon0 4128   suc csuc 4130   omcom 4341   `'ccnv 4372   dom cdm 4373   ran crn 4374   "cima 4376   Fun wfun 4924    Fn wfn 4925   -1-1->wf1 4927   -onto->wfo 4928   -1-1-onto->wf1o 4929   ` cfv 4930    ~~ cen 6250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-er 6137  df-en 6253
This theorem is referenced by: (None)
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