Theorem 1stinr 7077

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 7049	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inr = ( x e. _V |-> <. 1o , x >. ) )
3		opeq2 3781	. . . . 5 |- ( x = X -> <. 1o , x >. = <. 1o , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. 1o , x >. = <. 1o , X >. )
5		elex 2750	. . . 4 |- ( X e. V -> X e. _V )
6		1on 6426	. . . . 5 |- 1o e. On
7		opexg 4230	. . . . 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 5599	. . 3 |- ( X e. V -> (inr ` X ) = <. 1o , X >. )
10	9	fveq2d 5521	. 2 |- ( X e. V -> ( 1st ` (inr ` X ) ) = ( 1st ` <. 1o , X >. ) )
11		op1stg 6153	. . 3 |- ( ( 1o e. On /\ X e. V ) -> ( 1st ` <. 1o , X >. ) = 1o )
12	6, 11	mpan 424	. 2 |- ( X e. V -> ( 1st ` <. 1o , X >. ) = 1o )
13	10, 12	eqtrd 2210	1 |- ( X e. V -> ( 1st ` (inr ` X ) ) = 1o )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   <.cop 3597   |-> cmpt 4066   Oncon0 4365   ` cfv 5218   1stc1st 6141   1oc1o 6412  inrcinr 7047

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fv 5226  df-1st 6143  df-1o 6419  df-inr 7049

This theorem is referenced by:  djune  7079  updjudhcoinrg  7082