Theorem 1stinr 6961

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 6933	. . . . 5 |- inr = ( x e. _V |-> <. 1o , x >. )
2	1	a1i 9	. . . 4 |- ( X e. V -> inr = ( x e. _V |-> <. 1o , x >. ) )
3		opeq2 3706	. . . . 5 |- ( x = X -> <. 1o , x >. = <. 1o , X >. )
4	3	adantl 275	. . . 4 |- ( ( X e. V /\ x = X ) -> <. 1o , x >. = <. 1o , X >. )
5		elex 2697	. . . 4 |- ( X e. V -> X e. _V )
6		1on 6320	. . . . 5 |- 1o e. On
7		opexg 4150	. . . . 5 |- ( ( 1o e. On /\ X e. V ) -> <. 1o , X >. e. _V )
8	6, 7	mpan 420	. . . 4 |- ( X e. V -> <. 1o , X >. e. _V )
9	2, 4, 5, 8	fvmptd 5502	. . 3 |- ( X e. V -> (inr ` X ) = <. 1o , X >. )
10	9	fveq2d 5425	. 2 |- ( X e. V -> ( 1st ` (inr ` X ) ) = ( 1st ` <. 1o , X >. ) )
11		op1stg 6048	. . 3 |- ( ( 1o e. On /\ X e. V ) -> ( 1st ` <. 1o , X >. ) = 1o )
12	6, 11	mpan 420	. 2 |- ( X e. V -> ( 1st ` <. 1o , X >. ) = 1o )
13	10, 12	eqtrd 2172	1 |- ( X e. V -> ( 1st ` (inr ` X ) ) = 1o )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2686   <.cop 3530   |-> cmpt 3989   Oncon0 4285   ` cfv 5123   1stc1st 6036   1oc1o 6306  inrcinr 6931

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-1st 6038  df-1o 6313  df-inr 6933

This theorem is referenced by:  djune  6963  updjudhcoinrg  6966