Theorem 1stinl 7135

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 7108	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3806	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2771	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4157	. . . . 5 |- (/) e. _V
7		opexg 4258	. . . . 5 |- ( ( (/) e. _V /\ X e. V ) -> <. (/) , X >. e. _V )
8	6, 7	mpan 424	. . . 4 |- ( X e. V -> <. (/) , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5639	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5559	. 2 |- ( X e. V -> ( 1st ` (inl ` X ) ) = ( 1st ` <. (/) , X >. ) )
11		op1stg 6205	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 1st ` <. (/) , X >. ) = (/) )
12	6, 11	mpan 424	. 2 |- ( X e. V -> ( 1st ` <. (/) , X >. ) = (/) )
13	10, 12	eqtrd 2226	1 |- ( X e. V -> ( 1st ` (inl ` X ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   (/)c0 3447   <.cop 3622   |-> cmpt 4091   ` cfv 5255   1stc1st 6193  inlcinl 7106

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-1st 6195  df-inl 7108

This theorem is referenced by:  djune  7139  updjudhcoinlf  7141  subctctexmid  15561