Theorem bj-evalid 33028

Description: The evaluation at a set of the identity function is that set. (General form of ndxarg 15882.) The restriction to a set V is necessary since the argument of the function Slot A (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.)

Assertion

Ref	Expression
bj-evalid	|- ( ( V e. W /\ A e. V ) -> (Slot A ` ( _I |` V ) ) = A )

Proof of Theorem bj-evalid

Step	Hyp	Ref	Expression
1		resiexg 7102	. . 3 |- ( V e. W -> ( _I |` V ) e. _V )
2		bj-evalval 33027	. . 3 |- ( ( _I |` V ) e. _V -> (Slot A ` ( _I |` V ) ) = ( ( _I |` V ) ` A ) )
3	1, 2	syl 17	. 2 |- ( V e. W -> (Slot A ` ( _I |` V ) ) = ( ( _I |` V ) ` A ) )
4		fvresi 6439	. 2 |- ( A e. V -> ( ( _I |` V ) ` A ) = A )
5	3, 4	sylan9eq 2676	1 |- ( ( V e. W /\ A e. V ) -> (Slot A ` ( _I |` V ) ) = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   _I cid 5023   |` cres 5116   ` cfv 5888  Slot cslot 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861

This theorem is referenced by:  bj-ndxarg  33029  bj-evalidval  33031