Theorem 1stinl 7039

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 7012	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3759	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 275	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2737	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4109	. . . . 5 |- (/) e. _V
7		opexg 4206	. . . . 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 5567	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5490	. 2 |- ( X e. V -> ( 1st ` (inl ` X ) ) = ( 1st ` <. (/) , X >. ) )
11		op1stg 6118	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 1st ` <. (/) , X >. ) = (/) )
12	6, 11	mpan 421	. 2 |- ( X e. V -> ( 1st ` <. (/) , X >. ) = (/) )
13	10, 12	eqtrd 2198	1 |- ( X e. V -> ( 1st ` (inl ` X ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   (/)c0 3409   <.cop 3579   |-> cmpt 4043   ` cfv 5188   1stc1st 6106  inlcinl 7010

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-1st 6108  df-inl 7012

This theorem is referenced by:  djune  7043  updjudhcoinlf  7045  subctctexmid  13881