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Theorem eninl 6989
Description: Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
Assertion
Ref Expression
eninl  |-  ( A  e.  V  ->  (inl " A )  ~~  A
)

Proof of Theorem eninl
StepHypRef Expression
1 djulf1or 6948 . . . 4  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
2 f1oeng 6658 . . . 4  |-  ( ( A  e.  V  /\  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A ) )  ->  A  ~~  ( { (/) }  X.  A ) )
31, 2mpan2 422 . . 3  |-  ( A  e.  V  ->  A  ~~  ( { (/) }  X.  A ) )
4 df-ima 4559 . . . 4  |-  (inl " A )  =  ran  (inl  |`  A )
5 dff1o5 5383 . . . . . 6  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  <->  ( (inl  |`  A ) : A -1-1-> ( {
(/) }  X.  A
)  /\  ran  (inl  |`  A )  =  ( { (/) }  X.  A ) ) )
61, 5mpbi 144 . . . . 5  |-  ( (inl  |`  A ) : A -1-1-> ( { (/) }  X.  A
)  /\  ran  (inl  |`  A )  =  ( { (/) }  X.  A ) )
76simpri 112 . . . 4  |-  ran  (inl  |`  A )  =  ( { (/) }  X.  A
)
84, 7eqtri 2161 . . 3  |-  (inl " A )  =  ( { (/) }  X.  A
)
93, 8breqtrrdi 3977 . 2  |-  ( A  e.  V  ->  A  ~~  (inl " A ) )
109ensymd 6684 1  |-  ( A  e.  V  ->  (inl " A )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   (/)c0 3367   {csn 3531   class class class wbr 3936    X. cxp 4544   ran crn 4547    |` cres 4548   "cima 4549   -1-1->wf1 5127   -1-1-onto->wf1o 5129    ~~ cen 6639  inlcinl 6937
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-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  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-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  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-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  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-1st 6045  df-2nd 6046  df-er 6436  df-en 6642  df-inl 6939
This theorem is referenced by:  endjudisj  7082  djuen  7083
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