Theorem 1stinr 7274

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 7246	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inr = ( x e. _V |-> <. 1o , x >. ) )
3		opeq2 3863	. . . . 5 |- ( x = X -> <. 1o , x >. = <. 1o , X >. )
4	3	adantl 277	. . . 4 |- ( ( X e. V /\ x = X ) -> <. 1o , x >. = <. 1o , X >. )
5		elex 2814	. . . 4 |- ( X e. V -> X e. _V )
6		1on 6588	. . . . 5 |- 1o e. On
7		opexg 4320	. . . . 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 5727	. . 3 |- ( X e. V -> (inr ` X ) = <. 1o , X >. )
10	9	fveq2d 5643	. 2 |- ( X e. V -> ( 1st ` (inr ` X ) ) = ( 1st ` <. 1o , X >. ) )
11		op1stg 6312	. . 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 2264	1 |- ( X e. V -> ( 1st ` (inr ` X ) ) = 1o )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   |-> cmpt 4150   Oncon0 4460   ` cfv 5326   1stc1st 6300   1oc1o 6574  inrcinr 7244

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302  df-1o 6581  df-inr 7246

This theorem is referenced by:  djune  7276  updjudhcoinrg  7279