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Theorem phplem4on 6896
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 6774 . . . . 5  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
21biimpi 120 . . . 4  |-  ( suc 
A  ~~  suc  B  ->  E. f  f : suc  A -1-1-onto-> suc  B )
32adantl 277 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  suc  A  ~~  suc  B )  ->  E. f 
f : suc  A -1-1-onto-> suc  B )
4 f1of1 5479 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
54adantl 277 . . . . . . 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 4612 . . . . . . . . 9  |-  ( B  e.  om  ->  suc  B  e.  om )
7 nnon 4627 . . . . . . . . 9  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
86, 7syl 14 . . . . . . . 8  |-  ( B  e.  om  ->  suc  B  e.  On )
98ad3antlr 493 . . . . . . 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 4433 . . . . . . . 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 533 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  e.  On )
13 f1imaen2g 6820 . . . . . . 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 1250 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " A )  ~~  A
)
1514ensymd 6810 . . . . 5  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  ( f " A
) )
16 eloni 4393 . . . . . . . . 9  |-  ( A  e.  On  ->  Ord  A )
17 orddif 4564 . . . . . . . . 9  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1816, 17syl 14 . . . . . . . 8  |-  ( A  e.  On  ->  A  =  ( suc  A  \  { A } ) )
1918imaeq2d 4988 . . . . . . 7  |-  ( A  e.  On  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
2019ad3antrrr 492 . . . . . 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 5481 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
2221adantl 277 . . . . . . . . 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 4434 . . . . . . . . . 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 5596 . . . . . . . . 9  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2622, 24, 25syl2anc 411 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2726difeq2d 3268 . . . . . . 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 4998 . . . . . . . . . . 11  |-  ( f
" dom  f )  =  ran  f
2928eqcomi 2193 . . . . . . . . . 10  |-  ran  f  =  ( f " dom  f )
30 f1ofo 5487 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
31 forn 5460 . . . . . . . . . . 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 5484 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
3433imaeq2d 4988 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
3529, 32, 343eqtr3a 2246 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
3635difeq1d 3267 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
3736adantl 277 . . . . . . 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 5486 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3938simprbi 275 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
40 imadif 5315 . . . . . . . . 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 277 . . . . . . 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 2233 . . . . . 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 2222 . . . . 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 4044 . . . 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 534 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  B  e.  om )
47 fnfvelrn 5669 . . . . . . . 8  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
4822, 24, 47syl2anc 411 . . . . . . 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 2259 . . . . . . . . 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 277 . . . . . . 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 147 . . . . . 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 6885 . . . . . 6  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
5446, 52, 53syl2anc 411 . . . . 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 6810 . . . 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 6811 . . . 4  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
5745, 55, 56syl2anc 411 . . 3  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
583, 57exlimddv 1910 . 2  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  suc  A  ~~  suc  B )  ->  A  ~~  B )
5958ex 115 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160    \ cdif 3141    C_ wss 3144   {csn 3607   class class class wbr 4018   Ord word 4380   Oncon0 4381   suc csuc 4383   omcom 4607   `'ccnv 4643   dom cdm 4644   ran crn 4645   "cima 4647   Fun wfun 5229    Fn wfn 5230   -1-1->wf1 5232   -onto->wfo 5233   -1-1-onto->wf1o 5234   ` cfv 5235    ~~ cen 6765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-er 6560  df-en 6768
This theorem is referenced by: (None)
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