Theorem 2ndinl 7064

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 7036	. . . . 5 |- inl = ( x e. _V |-> <. (/) , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inl = ( x e. _V |-> <. (/) , x >. ) )
3		opeq2 3775	. . . . 5 |- ( x = X -> <. (/) , x >. = <. (/) , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. (/) , x >. = <. (/) , X >. )
5		elex 2746	. . . 4 |- ( X e. V -> X e. _V )
6		0ex 4125	. . . . 5 |- (/) e. _V
7		opexg 4222	. . . . 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 5589	. . 3 |- ( X e. V -> (inl ` X ) = <. (/) , X >. )
10	9	fveq2d 5511	. 2 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = ( 2nd ` <. (/) , X >. ) )
11		op2ndg 6142	. . 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 2208	1 |- ( X e. V -> ( 2nd ` (inl ` X ) ) = X )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   _Vcvv 2735   (/)c0 3420   <.cop 3592   |-> cmpt 4059   ` cfv 5208   2ndc2nd 6130  inlcinl 7034

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fv 5216  df-2nd 6132  df-inl 7036

This theorem is referenced by:  updjudhcoinlf  7069