Theorem 1stinr 7409

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 7381	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inr = ( x e. _V |-> <. 1o , x >. ) )
3		opeq2 3903	. . . . 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 6687	. . . . 5 |- 1o e. On
7		opexg 4366	. . . . 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 5783	. . 3 |- ( X e. V -> (inr ` X ) = <. 1o , X >. )
10	9	fveq2d 5697	. 2 |- ( X e. V -> ( 1st ` (inr ` X ) ) = ( 1st ` <. 1o , X >. ) )
11		op1stg 6377	. . 3 |- ( ( 1o e. On /\ X e. V ) -> ( 1st ` <. 1o , X >. ) = 1o )
12	6, 11	mpan 428	. 2 |- ( X e. V -> ( 1st ` <. 1o , X >. ) = 1o )
13	10, 12	eqtrd 2271	1 |- ( X e. V -> ( 1st ` (inr ` X ) ) = 1o )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3711   |-> cmpt 4190   Oncon0 4506   ` cfv 5375   1stc1st 6365   1oc1o 6673  inrcinr 7379

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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-1st 6367  df-1o 6680  df-inr 7381

This theorem is referenced by:  djune  7411  updjudhcoinrg  7414