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Theorem 2ndinl 6964
Description: The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
2ndinl  |-  ( X  e.  V  ->  ( 2nd `  (inl `  X
) )  =  X )

Proof of Theorem 2ndinl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-inl 6936 . . . . 5  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
21a1i 9 . . . 4  |-  ( X  e.  V  -> inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
)
3 opeq2 3710 . . . . 5  |-  ( x  =  X  ->  <. (/) ,  x >.  =  <. (/) ,  X >. )
43adantl 275 . . . 4  |-  ( ( X  e.  V  /\  x  =  X )  -> 
<. (/) ,  x >.  = 
<. (/) ,  X >. )
5 elex 2698 . . . 4  |-  ( X  e.  V  ->  X  e.  _V )
6 0ex 4059 . . . . 5  |-  (/)  e.  _V
7 opexg 4154 . . . . 5  |-  ( (
(/)  e.  _V  /\  X  e.  V )  ->  <. (/) ,  X >.  e.  _V )
86, 7mpan 421 . . . 4  |-  ( X  e.  V  ->  <. (/) ,  X >.  e.  _V )
92, 4, 5, 8fvmptd 5506 . . 3  |-  ( X  e.  V  ->  (inl `  X )  =  <. (/)
,  X >. )
109fveq2d 5429 . 2  |-  ( X  e.  V  ->  ( 2nd `  (inl `  X
) )  =  ( 2nd `  <. (/) ,  X >. ) )
11 op2ndg 6053 . . 3  |-  ( (
(/)  e.  _V  /\  X  e.  V )  ->  ( 2nd `  <. (/) ,  X >. )  =  X )
126, 11mpan 421 . 2  |-  ( X  e.  V  ->  ( 2nd `  <. (/) ,  X >. )  =  X )
1310, 12eqtrd 2173 1  |-  ( X  e.  V  ->  ( 2nd `  (inl `  X
) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    e. wcel 1481   _Vcvv 2687   (/)c0 3364   <.cop 3531    |-> cmpt 3993   ` cfv 5127   2ndc2nd 6041  inlcinl 6934
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-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-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-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-iota 5092  df-fun 5129  df-fv 5135  df-2nd 6043  df-inl 6936
This theorem is referenced by:  updjudhcoinlf  6969
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