Theorem orvcval 30519

Description: Value of the preimage mapping operator applied on a given random variable and constant. (Contributed by Thierry Arnoux, 19-Jan-2017.)

Hypotheses

Ref	Expression
orvcval.1	|- ( ph -> Fun X )
orvcval.2	|- ( ph -> X e. V )
orvcval.3	|- ( ph -> A e. W )

Assertion

Ref	Expression
orvcval	|- ( ph -> ( X∘RV/𝑐 R A ) = ( `' X " { y | y R A } ) )

Distinct variable groups:   y, A   y, R   y, X

Allowed substitution hints:   ph( y)   V( y)   W( y)

Proof of Theorem orvcval

Dummy variables x a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-orvc 30518	. . 3 |- ∘RV/𝑐 R = ( x e. { x | Fun x } , a e. _V |-> ( `' x " { y | y R a } ) )
2	1	a1i 11	. 2 |- ( ph -> ∘RV/𝑐 R = ( x e. { x | Fun x } , a e. _V |-> ( `' x " { y | y R a } ) ) )
3		simpl 473	. . . . 5 |- ( ( x = X /\ a = A ) -> x = X )
4	3	cnveqd 5298	. . . 4 |- ( ( x = X /\ a = A ) -> `' x = `' X )
5		simpr 477	. . . . . 6 |- ( ( x = X /\ a = A ) -> a = A )
6	5	breq2d 4665	. . . . 5 |- ( ( x = X /\ a = A ) -> ( y R a <-> y R A ) )
7	6	abbidv 2741	. . . 4 |- ( ( x = X /\ a = A ) -> { y | y R a } = { y | y R A } )
8	4, 7	imaeq12d 5467	. . 3 |- ( ( x = X /\ a = A ) -> ( `' x " { y | y R a } ) = ( `' X " { y | y R A } ) )
9	8	adantl 482	. 2 |- ( ( ph /\ ( x = X /\ a = A ) ) -> ( `' x " { y | y R a } ) = ( `' X " { y | y R A } ) )
10		orvcval.1	. . 3 |- ( ph -> Fun X )
11		orvcval.2	. . . 4 |- ( ph -> X e. V )
12		funeq 5908	. . . . 5 |- ( x = X -> ( Fun x <-> Fun X ) )
13	12	elabg 3351	. . . 4 |- ( X e. V -> ( X e. { x | Fun x } <-> Fun X ) )
14	11, 13	syl 17	. . 3 |- ( ph -> ( X e. { x | Fun x } <-> Fun X ) )
15	10, 14	mpbird 247	. 2 |- ( ph -> X e. { x | Fun x } )
16		orvcval.3	. . 3 |- ( ph -> A e. W )
17		elex 3212	. . 3 |- ( A e. W -> A e. _V )
18	16, 17	syl 17	. 2 |- ( ph -> A e. _V )
19		cnvexg 7112	. . 3 |- ( X e. V -> `' X e. _V )
20		imaexg 7103	. . 3 |- ( `' X e. _V -> ( `' X " { y | y R A } ) e. _V )
21	11, 19, 20	3syl 18	. 2 |- ( ph -> ( `' X " { y | y R A } ) e. _V )
22	2, 9, 15, 18, 21	ovmpt2d 6788	1 |- ( ph -> ( X∘RV/𝑐 R A ) = ( `' X " { y | y R A } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   "cima 5117   Fun wfun 5882  (class class class)co 6650   |-> cmpt2 6652  ∘_RV/𝑐corvc 30517

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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-orvc 30518

This theorem is referenced by:  orvcval2  30520  orvcval4  30522