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Theorem 1stinl 6925
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 6898 . . . . 5  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
21a1i 9 . . . 4  |-  ( X  e.  V  -> inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
)
3 opeq2 3674 . . . . 5  |-  ( x  =  X  ->  <. (/) ,  x >.  =  <. (/) ,  X >. )
43adantl 273 . . . 4  |-  ( ( X  e.  V  /\  x  =  X )  -> 
<. (/) ,  x >.  = 
<. (/) ,  X >. )
5 elex 2669 . . . 4  |-  ( X  e.  V  ->  X  e.  _V )
6 0ex 4023 . . . . 5  |-  (/)  e.  _V
7 opexg 4118 . . . . 5  |-  ( (
(/)  e.  _V  /\  X  e.  V )  ->  <. (/) ,  X >.  e.  _V )
86, 7mpan 418 . . . 4  |-  ( X  e.  V  ->  <. (/) ,  X >.  e.  _V )
92, 4, 5, 8fvmptd 5468 . . 3  |-  ( X  e.  V  ->  (inl `  X )  =  <. (/)
,  X >. )
109fveq2d 5391 . 2  |-  ( X  e.  V  ->  ( 1st `  (inl `  X
) )  =  ( 1st `  <. (/) ,  X >. ) )
11 op1stg 6014 . . 3  |-  ( (
(/)  e.  _V  /\  X  e.  V )  ->  ( 1st `  <. (/) ,  X >. )  =  (/) )
126, 11mpan 418 . 2  |-  ( X  e.  V  ->  ( 1st `  <. (/) ,  X >. )  =  (/) )
1310, 12eqtrd 2148 1  |-  ( X  e.  V  ->  ( 1st `  (inl `  X
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2658   (/)c0 3331   <.cop 3498    |-> cmpt 3957   ` cfv 5091   1stc1st 6002  inlcinl 6896
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fv 5099  df-1st 6004  df-inl 6898
This theorem is referenced by:  djune  6929  updjudhcoinlf  6931  subctctexmid  13030
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