Theorem 2ndinr 7336

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 7307	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inr = ( x e. _V |-> <. 1o , x >. ) )
3		opeq2 3868	. . . . 5 |- ( x = X -> <. 1o , x >. = <. 1o , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. 1o , x >. = <. 1o , X >. )
5		elex 2815	. . . 4 |- ( X e. V -> X e. _V )
6		1on 6632	. . . . 5 |- 1o e. On
7		opexg 4326	. . . . 5 |- ( ( 1o e. On /\ X e. V ) -> <. 1o , X >. e. _V )
8	6, 7	mpan 424	. . . 4 |- ( X e. V -> <. 1o , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5736	. . 3 |- ( X e. V -> (inr ` X ) = <. 1o , X >. )
10	9	fveq2d 5652	. 2 |- ( X e. V -> ( 2nd ` (inr ` X ) ) = ( 2nd ` <. 1o , X >. ) )
11		op2ndg 6323	. . 3 |- ( ( 1o e. On /\ X e. V ) -> ( 2nd ` <. 1o , X >. ) = X )
12	6, 11	mpan 424	. 2 |- ( X e. V -> ( 2nd ` <. 1o , X >. ) = X )
13	10, 12	eqtrd 2264	1 |- ( X e. V -> ( 2nd ` (inr ` X ) ) = X )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   <.cop 3676   |-> cmpt 4155   Oncon0 4466   ` cfv 5333   2ndc2nd 6311   1oc1o 6618  inrcinr 7305

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-2nd 6313  df-1o 6625  df-inr 7307

This theorem is referenced by:  updjudhcoinrg  7340