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Theorem phplem4on 6857
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 6737 . . . . 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 5452 . . . . . . . 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 4588 . . . . . . . . 9  |-  ( B  e.  om  ->  suc  B  e.  om )
7 nnon 4603 . . . . . . . . 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 4409 . . . . . . . 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 6783 . . . . . . 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 1239 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  ( f " A )  ~~  A
)
1514ensymd 6773 . . . . 5  |-  ( ( ( ( A  e.  On  /\  B  e. 
om )  /\  suc  A 
~~  suc  B )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  ( f " A
) )
16 eloni 4369 . . . . . . . . 9  |-  ( A  e.  On  ->  Ord  A )
17 orddif 4540 . . . . . . . . 9  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1816, 17syl 14 . . . . . . . 8  |-  ( A  e.  On  ->  A  =  ( suc  A  \  { A } ) )
1918imaeq2d 4963 . . . . . . 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 5454 . . . . . . . . . 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 4410 . . . . . . . . . 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 5567 . . . . . . . . 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 3251 . . . . . . 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 4973 . . . . . . . . . . 11  |-  ( f
" dom  f )  =  ran  f
2928eqcomi 2179 . . . . . . . . . 10  |-  ran  f  =  ( f " dom  f )
30 f1ofo 5460 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
31 forn 5433 . . . . . . . . . . 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 5457 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
3433imaeq2d 4963 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
3529, 32, 343eqtr3a 2232 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
3635difeq1d 3250 . . . . . . . 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 5459 . . . . . . . . . 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 5288 . . . . . . . . 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 2219 . . . . . 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 2208 . . . . 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 4024 . . . 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 5640 . . . . . . . 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 2245 . . . . . . . . 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 6846 . . . . . 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 6773 . . . 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 6774 . . . 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 1896 . 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 1353   E.wex 1490    e. wcel 2146    \ cdif 3124    C_ wss 3127   {csn 3589   class class class wbr 3998   Ord word 4356   Oncon0 4357   suc csuc 4359   omcom 4583   `'ccnv 4619   dom cdm 4620   ran crn 4621   "cima 4623   Fun wfun 5202    Fn wfn 5203   -1-1->wf1 5205   -onto->wfo 5206   -1-1-onto->wf1o 5207   ` cfv 5208    ~~ cen 6728
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-er 6525  df-en 6731
This theorem is referenced by: (None)
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