Theorem 1stinr 7021

Description: The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
1stinr	|- ( X e. V -> ( 1st ` (inr ` X ) ) = 1o )

Proof of Theorem 1stinr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inr 6993	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inr = ( x e. _V |-> <. 1o , x >. ) )
3		opeq2 3743	. . . . 5 |- ( x = X -> <. 1o , x >. = <. 1o , X >. )
4	3	adantl 275	. . . 4 |- ( ( X e. V /\ x = X ) -> <. 1o , x >. = <. 1o , X >. )
5		elex 2723	. . . 4 |- ( X e. V -> X e. _V )
6		1on 6371	. . . . 5 |- 1o e. On
7		opexg 4189	. . . . 5 |- ( ( 1o e. On /\ X e. V ) -> <. 1o , X >. e. _V )
8	6, 7	mpan 421	. . . 4 |- ( X e. V -> <. 1o , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5550	. . 3 |- ( X e. V -> (inr ` X ) = <. 1o , X >. )
10	9	fveq2d 5473	. 2 |- ( X e. V -> ( 1st ` (inr ` X ) ) = ( 1st ` <. 1o , X >. ) )
11		op1stg 6099	. . 3 |- ( ( 1o e. On /\ X e. V ) -> ( 1st ` <. 1o , X >. ) = 1o )
12	6, 11	mpan 421	. 2 |- ( X e. V -> ( 1st ` <. 1o , X >. ) = 1o )
13	10, 12	eqtrd 2190	1 |- ( X e. V -> ( 1st ` (inr ` X ) ) = 1o )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   _Vcvv 2712   <.cop 3563   |-> cmpt 4026   Oncon0 4324   ` cfv 5171   1stc1st 6087   1oc1o 6357  inrcinr 6991

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-suc 4332  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fv 5179  df-1st 6089  df-1o 6364  df-inr 6993

This theorem is referenced by:  djune  7023  updjudhcoinrg  7026