Theorem 1stinl 7367

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 7340	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3886	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2827	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4239	. . . . 5 |- (/) e. _V
7		opexg 4346	. . . . 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 5760	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5676	. 2 |- ( X e. V -> ( 1st ` (inl ` X ) ) = ( 1st ` <. (/) , X >. ) )
11		op1stg 6346	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 1st ` <. (/) , X >. ) = (/) )
12	6, 11	mpan 424	. 2 |- ( X e. V -> ( 1st ` <. (/) , X >. ) = (/) )
13	10, 12	eqtrd 2267	1 |- ( X e. V -> ( 1st ` (inl ` X ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   (/)c0 3510   <.cop 3694   |-> cmpt 4173   ` cfv 5354   1stc1st 6334  inlcinl 7338

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fv 5362  df-1st 6336  df-inl 7340

This theorem is referenced by:  djune  7371  updjudhcoinlf  7373  subctctexmid  16791