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Theorem 1stinl 7066
Description: The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
1stinl  |-  ( X  e.  V  ->  ( 1st `  (inl `  X
) )  =  (/) )

Proof of Theorem 1stinl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-inl 7039 . . . . 5  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
21a1i 9 . . . 4  |-  ( X  e.  V  -> inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
)
3 opeq2 3777 . . . . 5  |-  ( x  =  X  ->  <. (/) ,  x >.  =  <. (/) ,  X >. )
43adantl 277 . . . 4  |-  ( ( X  e.  V  /\  x  =  X )  -> 
<. (/) ,  x >.  = 
<. (/) ,  X >. )
5 elex 2748 . . . 4  |-  ( X  e.  V  ->  X  e.  _V )
6 0ex 4127 . . . . 5  |-  (/)  e.  _V
7 opexg 4224 . . . . 5  |-  ( (
(/)  e.  _V  /\  X  e.  V )  ->  <. (/) ,  X >.  e.  _V )
86, 7mpan 424 . . . 4  |-  ( X  e.  V  ->  <. (/) ,  X >.  e.  _V )
92, 4, 5, 8fvmptd 5592 . . 3  |-  ( X  e.  V  ->  (inl `  X )  =  <. (/)
,  X >. )
109fveq2d 5514 . 2  |-  ( X  e.  V  ->  ( 1st `  (inl `  X
) )  =  ( 1st `  <. (/) ,  X >. ) )
11 op1stg 6144 . . 3  |-  ( (
(/)  e.  _V  /\  X  e.  V )  ->  ( 1st `  <. (/) ,  X >. )  =  (/) )
126, 11mpan 424 . 2  |-  ( X  e.  V  ->  ( 1st `  <. (/) ,  X >. )  =  (/) )
1310, 12eqtrd 2210 1  |-  ( X  e.  V  ->  ( 1st `  (inl `  X
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737   (/)c0 3422   <.cop 3594    |-> cmpt 4061   ` cfv 5211   1stc1st 6132  inlcinl 7037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fv 5219  df-1st 6134  df-inl 7039
This theorem is referenced by:  djune  7070  updjudhcoinlf  7072  subctctexmid  14373
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