Theorem 1stinl 6967

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.

Step	Hyp	Ref	Expression
1		df-inl 6940	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3714	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 275	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2700	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4063	. . . . 5 |- (/) e. _V
7		opexg 4158	. . . . 5 |- ( ( (/) e. _V /\ X e. V ) -> <. (/) , X >. e. _V )
8	6, 7	mpan 421	. . . 4 |- ( X e. V -> <. (/) , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5510	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5433	. 2 |- ( X e. V -> ( 1st ` (inl ` X ) ) = ( 1st ` <. (/) , X >. ) )
11		op1stg 6056	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 1st ` <. (/) , X >. ) = (/) )
12	6, 11	mpan 421	. 2 |- ( X e. V -> ( 1st ` <. (/) , X >. ) = (/) )
13	10, 12	eqtrd 2173	1 |- ( X e. V -> ( 1st ` (inl ` X ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   _Vcvv 2689   (/)c0 3368   <.cop 3535   |-> cmpt 3997   ` cfv 5131   1stc1st 6044  inlcinl 6938

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139  df-1st 6046  df-inl 6940

This theorem is referenced by:  djune  6971  updjudhcoinlf  6973  subctctexmid  13369