Theorem 2ndinr 7407

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

Assertion

Ref	Expression
2ndinr	|- ( X e. V -> ( 2nd ` (inr ` X ) ) = X )

Proof of Theorem 2ndinr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inr 7378	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inr = ( x e. _V |-> <. 1o , x >. ) )
3		opeq2 3900	. . . . 5 |- ( x = X -> <. 1o , x >. = <. 1o , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. 1o , x >. = <. 1o , X >. )
5		elex 2833	. . . 4 |- ( X e. V -> X e. _V )
6		1on 6684	. . . . 5 |- 1o e. On
7		opexg 4363	. . . . 5 |- ( ( 1o e. On /\ X e. V ) -> <. 1o , X >. e. _V )
8	6, 7	mpan 428	. . . 4 |- ( X e. V -> <. 1o , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5780	. . 3 |- ( X e. V -> (inr ` X ) = <. 1o , X >. )
10	9	fveq2d 5694	. 2 |- ( X e. V -> ( 2nd ` (inr ` X ) ) = ( 2nd ` <. 1o , X >. ) )
11		op2ndg 6375	. . 3 |- ( ( 1o e. On /\ X e. V ) -> ( 2nd ` <. 1o , X >. ) = X )
12	6, 11	mpan 428	. 2 |- ( X e. V -> ( 2nd ` <. 1o , X >. ) = X )
13	10, 12	eqtrd 2271	1 |- ( X e. V -> ( 2nd ` (inr ` X ) ) = X )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3708   |-> cmpt 4187   Oncon0 4503   ` cfv 5372   2ndc2nd 6363   1oc1o 6670  inrcinr 7376

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-2nd 6365  df-1o 6677  df-inr 7378

This theorem is referenced by:  updjudhcoinrg  7411