Theorem 1stinl 7149

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 7122	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3810	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2774	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4161	. . . . 5 |- (/) e. _V
7		opexg 4262	. . . . 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 5645	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5565	. 2 |- ( X e. V -> ( 1st ` (inl ` X ) ) = ( 1st ` <. (/) , X >. ) )
11		op1stg 6217	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 1st ` <. (/) , X >. ) = (/) )
12	6, 11	mpan 424	. 2 |- ( X e. V -> ( 1st ` <. (/) , X >. ) = (/) )
13	10, 12	eqtrd 2229	1 |- ( X e. V -> ( 1st ` (inl ` X ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   (/)c0 3451   <.cop 3626   |-> cmpt 4095   ` cfv 5259   1stc1st 6205  inlcinl 7120

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fv 5267  df-1st 6207  df-inl 7122

This theorem is referenced by:  djune  7153  updjudhcoinlf  7155  subctctexmid  15731