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Theorem 1stinr 6765
Description: The first component of the value of a right injection is 
1o. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
1stinr  |-  ( X  e.  V  ->  ( 1st `  (inr `  X
) )  =  1o )

Proof of Theorem 1stinr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-inr 6738 . . . . 5  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
21a1i 9 . . . 4  |-  ( X  e.  V  -> inr  =  ( x  e.  _V  |->  <. 1o ,  x >. ) )
3 opeq2 3623 . . . . 5  |-  ( x  =  X  ->  <. 1o ,  x >.  =  <. 1o ,  X >. )
43adantl 271 . . . 4  |-  ( ( X  e.  V  /\  x  =  X )  -> 
<. 1o ,  x >.  = 
<. 1o ,  X >. )
5 elex 2630 . . . 4  |-  ( X  e.  V  ->  X  e.  _V )
6 1on 6188 . . . . 5  |-  1o  e.  On
7 opexg 4055 . . . . 5  |-  ( ( 1o  e.  On  /\  X  e.  V )  -> 
<. 1o ,  X >.  e. 
_V )
86, 7mpan 415 . . . 4  |-  ( X  e.  V  ->  <. 1o ,  X >.  e.  _V )
92, 4, 5, 8fvmptd 5385 . . 3  |-  ( X  e.  V  ->  (inr `  X )  =  <. 1o ,  X >. )
109fveq2d 5309 . 2  |-  ( X  e.  V  ->  ( 1st `  (inr `  X
) )  =  ( 1st `  <. 1o ,  X >. ) )
11 op1stg 5921 . . 3  |-  ( ( 1o  e.  On  /\  X  e.  V )  ->  ( 1st `  <. 1o ,  X >. )  =  1o )
126, 11mpan 415 . 2  |-  ( X  e.  V  ->  ( 1st `  <. 1o ,  X >. )  =  1o )
1310, 12eqtrd 2120 1  |-  ( X  e.  V  ->  ( 1st `  (inr `  X
) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   <.cop 3449    |-> cmpt 3899   Oncon0 4190   ` cfv 5015   1stc1st 5909   1oc1o 6174  inrcinr 6736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-1o 6181  df-inr 6738
This theorem is referenced by:  djune  6767  updjudhcoinrg  6770
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