Theorem 2ndinl 7238

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 7210	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3857	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2811	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4210	. . . . 5 |- (/) e. _V
7		opexg 4313	. . . . 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 5714	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5630	. 2 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = ( 2nd ` <. (/) , X >. ) )
11		op2ndg 6295	. . 3 |- ( ( (/) e. _V /\ X e. V ) -> ( 2nd ` <. (/) , X >. ) = X )
12	6, 11	mpan 424	. 2 |- ( X e. V -> ( 2nd ` <. (/) , X >. ) = X )
13	10, 12	eqtrd 2262	1 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = X )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   (/)c0 3491   <.cop 3669   |-> cmpt 4144   ` cfv 5317   2ndc2nd 6283  inlcinl 7208

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fv 5325  df-2nd 6285  df-inl 7210

This theorem is referenced by:  updjudhcoinlf  7243