Theorem 2ndinl 6909

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 6881	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3670	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 273	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2666	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4013	. . . . 5 |- (/) e. _V
7		opexg 4108	. . . . 5 |- ( ( (/) e. _V /\ X e. V ) -> <. (/) , X >. e. _V )
8	6, 7	mpan 418	. . . 4 |- ( X e. V -> <. (/) , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5454	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5377	. 2 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = ( 2nd ` <. (/) , X >. ) )
11		op2ndg 6000	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 2nd ` <. (/) , X >. ) = X )
12	6, 11	mpan 418	. 2 |- ( X e. V -> ( 2nd ` <. (/) , X >. ) = X )
13	10, 12	eqtrd 2145	1 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = X )

Syntax hints:   -> wi 4   = wceq 1312   e. wcel 1461   _Vcvv 2655   (/)c0 3327   <.cop 3494   |-> cmpt 3947   ` cfv 5079   2ndc2nd 5988  inlcinl 6879

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-iota 5044  df-fun 5081  df-fv 5087  df-2nd 5990  df-inl 6881

This theorem is referenced by:  updjudhcoinlf  6914