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Theorem phplem4on 6768
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 6648 . . . . 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 275 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  suc  A  ~~  suc  B )  ->  E. f 
f : suc  A -1-1-onto-> suc  B )
4 f1of1 5373 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
54adantl 275 . . . . . . 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 4516 . . . . . . . . 9  |-  ( B  e.  om  ->  suc  B  e.  om )
7 nnon 4530 . . . . . . . . 9  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
86, 7syl 14 . . . . . . . 8  |-  ( B  e.  om  ->  suc  B  e.  On )
98ad3antlr 485 . . . . . . 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 4344 . . . . . . . 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 523 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  e.  On )
13 f1imaen2g 6694 . . . . . . 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 1218 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " A )  ~~  A
)
1514ensymd 6684 . . . . 5  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  ( f " A
) )
16 eloni 4304 . . . . . . . . 9  |-  ( A  e.  On  ->  Ord  A )
17 orddif 4469 . . . . . . . . 9  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1816, 17syl 14 . . . . . . . 8  |-  ( A  e.  On  ->  A  =  ( suc  A  \  { A } ) )
1918imaeq2d 4888 . . . . . . 7  |-  ( A  e.  On  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
2019ad3antrrr 484 . . . . . 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 5375 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
2221adantl 275 . . . . . . . . 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 4345 . . . . . . . . . 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 5487 . . . . . . . . 9  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2622, 24, 25syl2anc 409 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2726difeq2d 3198 . . . . . . 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 4898 . . . . . . . . . . 11  |-  ( f
" dom  f )  =  ran  f
2928eqcomi 2144 . . . . . . . . . 10  |-  ran  f  =  ( f " dom  f )
30 f1ofo 5381 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
31 forn 5355 . . . . . . . . . . 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 5378 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
3433imaeq2d 4888 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
3529, 32, 343eqtr3a 2197 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
3635difeq1d 3197 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
3736adantl 275 . . . . . . 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 5380 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3938simprbi 273 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
40 imadif 5210 . . . . . . . . 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 275 . . . . . . 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 2184 . . . . . 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 2173 . . . . 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 3961 . . . 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 524 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  B  e.  om )
47 fnfvelrn 5559 . . . . . . . 8  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
4822, 24, 47syl2anc 409 . . . . . . 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 2210 . . . . . . . . 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 275 . . . . . . 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 6757 . . . . . 6  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
5446, 52, 53syl2anc 409 . . . . 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 6684 . . . 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 6685 . . . 4  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
5745, 55, 56syl2anc 409 . . 3  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
583, 57exlimddv 1871 . 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 1332   E.wex 1469    e. wcel 1481    \ cdif 3072    C_ wss 3075   {csn 3531   class class class wbr 3936   Ord word 4291   Oncon0 4292   suc csuc 4294   omcom 4511   `'ccnv 4545   dom cdm 4546   ran crn 4547   "cima 4549   Fun wfun 5124    Fn wfn 5125   -1-1->wf1 5127   -onto->wfo 5128   -1-1-onto->wf1o 5129   ` cfv 5130    ~~ cen 6639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-er 6436  df-en 6642
This theorem is referenced by: (None)
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