Theorem 2ndinl 7031

Description: The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
2ndinl	|- ( X e. V -> ( 2nd ` (inl ` X ) ) = X )

Proof of Theorem 2ndinl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inl 7003	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3753	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 275	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2732	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4103	. . . . 5 |- (/) e. _V
7		opexg 4200	. . . . 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 5561	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5484	. 2 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = ( 2nd ` <. (/) , X >. ) )
11		op2ndg 6111	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 2nd ` <. (/) , X >. ) = X )
12	6, 11	mpan 421	. 2 |- ( X e. V -> ( 2nd ` <. (/) , X >. ) = X )
13	10, 12	eqtrd 2197	1 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = X )

Syntax hints:   -> wi 4   = wceq 1342   e. wcel 2135   _Vcvv 2721   (/)c0 3404   <.cop 3573   |-> cmpt 4037   ` cfv 5182   2ndc2nd 6099  inlcinl 7001

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fv 5190  df-2nd 6101  df-inl 7003

This theorem is referenced by:  updjudhcoinlf  7036