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Theorem eninr 6933
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 6892 . . . 4  |-  (inr  |`  A ) : A -1-1-onto-> ( { 1o }  X.  A )
2 f1oeng 6603 . . . 4  |-  ( ( A  e.  V  /\  (inr  |`  A ) : A -1-1-onto-> ( { 1o }  X.  A ) )  ->  A  ~~  ( { 1o }  X.  A ) )
31, 2mpan2 419 . . 3  |-  ( A  e.  V  ->  A  ~~  ( { 1o }  X.  A ) )
4 df-ima 4510 . . . 4  |-  (inr " A )  =  ran  (inr  |`  A )
5 dff1o5 5330 . . . . . 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 2133 . . 3  |-  (inr " A )  =  ( { 1o }  X.  A )
93, 8syl6breqr 3933 . 2  |-  ( A  e.  V  ->  A  ~~  (inr " A ) )
109ensymd 6629 1  |-  ( A  e.  V  ->  (inr " A )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1312    e. wcel 1461   {csn 3491   class class class wbr 3893    X. cxp 4495   ran crn 4498    |` cres 4499   "cima 4500   -1-1->wf1 5076   -1-1-onto->wf1o 5078   1oc1o 6258    ~~ cen 6584  inrcinr 6881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-1st 5990  df-2nd 5991  df-1o 6265  df-er 6381  df-en 6587  df-inr 6883
This theorem is referenced by:  endjudisj  7011  djuen  7012
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