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Theorem eninr 6987
Description: Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
Assertion
Ref Expression
eninr  |-  ( A  e.  V  ->  (inr " A )  ~~  A
)

Proof of Theorem eninr
StepHypRef Expression
1 djurf1or 6946 . . . 4  |-  (inr  |`  A ) : A -1-1-onto-> ( { 1o }  X.  A )
2 f1oeng 6655 . . . 4  |-  ( ( A  e.  V  /\  (inr  |`  A ) : A -1-1-onto-> ( { 1o }  X.  A ) )  ->  A  ~~  ( { 1o }  X.  A ) )
31, 2mpan2 422 . . 3  |-  ( A  e.  V  ->  A  ~~  ( { 1o }  X.  A ) )
4 df-ima 4556 . . . 4  |-  (inr " A )  =  ran  (inr  |`  A )
5 dff1o5 5380 . . . . . 6  |-  ( (inr  |`  A ) : A -1-1-onto-> ( { 1o }  X.  A
)  <->  ( (inr  |`  A ) : A -1-1-> ( { 1o }  X.  A
)  /\  ran  (inr  |`  A )  =  ( { 1o }  X.  A ) ) )
61, 5mpbi 144 . . . . 5  |-  ( (inr  |`  A ) : A -1-1-> ( { 1o }  X.  A )  /\  ran  (inr  |`  A )  =  ( { 1o }  X.  A ) )
76simpri 112 . . . 4  |-  ran  (inr  |`  A )  =  ( { 1o }  X.  A )
84, 7eqtri 2161 . . 3  |-  (inr " A )  =  ( { 1o }  X.  A )
93, 8breqtrrdi 3974 . 2  |-  ( A  e.  V  ->  A  ~~  (inr " A ) )
109ensymd 6681 1  |-  ( A  e.  V  ->  (inr " A )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   {csn 3528   class class class wbr 3933    X. cxp 4541   ran crn 4544    |` cres 4545   "cima 4546   -1-1->wf1 5124   -1-1-onto->wf1o 5126   1oc1o 6310    ~~ cen 6636  inrcinr 6935
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 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359
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 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-1st 6042  df-2nd 6043  df-1o 6317  df-er 6433  df-en 6639  df-inr 6937
This theorem is referenced by:  endjudisj  7079  djuen  7080
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