ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1stinr Unicode version

Theorem 1stinr 7021
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 6993 . . . . 5  |- inr  =  ( x  e.  _V  |->  <. 1o ,  x >. )
21a1i 9 . . . 4  |-  ( X  e.  V  -> inr  =  ( x  e.  _V  |->  <. 1o ,  x >. ) )
3 opeq2 3743 . . . . 5  |-  ( x  =  X  ->  <. 1o ,  x >.  =  <. 1o ,  X >. )
43adantl 275 . . . 4  |-  ( ( X  e.  V  /\  x  =  X )  -> 
<. 1o ,  x >.  = 
<. 1o ,  X >. )
5 elex 2723 . . . 4  |-  ( X  e.  V  ->  X  e.  _V )
6 1on 6371 . . . . 5  |-  1o  e.  On
7 opexg 4189 . . . . 5  |-  ( ( 1o  e.  On  /\  X  e.  V )  -> 
<. 1o ,  X >.  e. 
_V )
86, 7mpan 421 . . . 4  |-  ( X  e.  V  ->  <. 1o ,  X >.  e.  _V )
92, 4, 5, 8fvmptd 5550 . . 3  |-  ( X  e.  V  ->  (inr `  X )  =  <. 1o ,  X >. )
109fveq2d 5473 . 2  |-  ( X  e.  V  ->  ( 1st `  (inr `  X
) )  =  ( 1st `  <. 1o ,  X >. ) )
11 op1stg 6099 . . 3  |-  ( ( 1o  e.  On  /\  X  e.  V )  ->  ( 1st `  <. 1o ,  X >. )  =  1o )
126, 11mpan 421 . 2  |-  ( X  e.  V  ->  ( 1st `  <. 1o ,  X >. )  =  1o )
1310, 12eqtrd 2190 1  |-  ( X  e.  V  ->  ( 1st `  (inr `  X
) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128   _Vcvv 2712   <.cop 3563    |-> cmpt 4026   Oncon0 4324   ` cfv 5171   1stc1st 6087   1oc1o 6357  inrcinr 6991
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-suc 4332  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fv 5179  df-1st 6089  df-1o 6364  df-inr 6993
This theorem is referenced by:  djune  7023  updjudhcoinrg  7026
  Copyright terms: Public domain W3C validator