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Theorem eninl 7070
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 7029 . . . 4  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
2 f1oeng 6731 . . . 4  |-  ( ( A  e.  V  /\  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A ) )  ->  A  ~~  ( { (/) }  X.  A ) )
31, 2mpan2 423 . . 3  |-  ( A  e.  V  ->  A  ~~  ( { (/) }  X.  A ) )
4 df-ima 4622 . . . 4  |-  (inl " A )  =  ran  (inl  |`  A )
5 dff1o5 5449 . . . . . 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 2191 . . 3  |-  (inl " A )  =  ( { (/) }  X.  A
)
93, 8breqtrrdi 4029 . 2  |-  ( A  e.  V  ->  A  ~~  (inl " A ) )
109ensymd 6757 1  |-  ( A  e.  V  ->  (inl " A )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   (/)c0 3414   {csn 3581   class class class wbr 3987    X. cxp 4607   ran crn 4610    |` cres 4611   "cima 4612   -1-1->wf1 5193   -1-1-onto->wf1o 5195    ~~ cen 6712  inlcinl 7018
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6116  df-2nd 6117  df-er 6509  df-en 6715  df-inl 7020
This theorem is referenced by:  endjudisj  7174  djuen  7175
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