Theorem 2ndinl 6960

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 6932	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3706	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 275	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2697	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4055	. . . . 5 |- (/) e. _V
7		opexg 4150	. . . . 5 |- ( ( (/) e. _V /\ X e. V ) -> <. (/) , X >. e. _V )
8	6, 7	mpan 420	. . . 4 |- ( X e. V -> <. (/) , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5502	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5425	. 2 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = ( 2nd ` <. (/) , X >. ) )
11		op2ndg 6049	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 2nd ` <. (/) , X >. ) = X )
12	6, 11	mpan 420	. 2 |- ( X e. V -> ( 2nd ` <. (/) , X >. ) = X )
13	10, 12	eqtrd 2172	1 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = X )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2686   (/)c0 3363   <.cop 3530   |-> cmpt 3989   ` cfv 5123   2ndc2nd 6037  inlcinl 6930

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-2nd 6039  df-inl 6932

This theorem is referenced by:  updjudhcoinlf  6965