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Theorem 2ndinr 6930
Description: The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
2ndinr  |-  ( X  e.  V  ->  ( 2nd `  (inr `  X
) )  =  X )

Proof of Theorem 2ndinr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-inr 6901 . . . . 5  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
21a1i 9 . . . 4  |-  ( X  e.  V  -> inr  =  ( x  e.  _V  |->  <. 1o ,  x >. ) )
3 opeq2 3676 . . . . 5  |-  ( x  =  X  ->  <. 1o ,  x >.  =  <. 1o ,  X >. )
43adantl 275 . . . 4  |-  ( ( X  e.  V  /\  x  =  X )  -> 
<. 1o ,  x >.  = 
<. 1o ,  X >. )
5 elex 2671 . . . 4  |-  ( X  e.  V  ->  X  e.  _V )
6 1on 6288 . . . . 5  |-  1o  e.  On
7 opexg 4120 . . . . 5  |-  ( ( 1o  e.  On  /\  X  e.  V )  -> 
<. 1o ,  X >.  e. 
_V )
86, 7mpan 420 . . . 4  |-  ( X  e.  V  ->  <. 1o ,  X >.  e.  _V )
92, 4, 5, 8fvmptd 5470 . . 3  |-  ( X  e.  V  ->  (inr `  X )  =  <. 1o ,  X >. )
109fveq2d 5393 . 2  |-  ( X  e.  V  ->  ( 2nd `  (inr `  X
) )  =  ( 2nd `  <. 1o ,  X >. ) )
11 op2ndg 6017 . . 3  |-  ( ( 1o  e.  On  /\  X  e.  V )  ->  ( 2nd `  <. 1o ,  X >. )  =  X )
126, 11mpan 420 . 2  |-  ( X  e.  V  ->  ( 2nd `  <. 1o ,  X >. )  =  X )
1310, 12eqtrd 2150 1  |-  ( X  e.  V  ->  ( 2nd `  (inr `  X
) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   _Vcvv 2660   <.cop 3500    |-> cmpt 3959   Oncon0 4255   ` cfv 5093   2ndc2nd 6005   1oc1o 6274  inrcinr 6899
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-2nd 6007  df-1o 6281  df-inr 6901
This theorem is referenced by:  updjudhcoinrg  6934
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