Theorem elorvc 30521

Description: Elementhood of a preimage. (Contributed by Thierry Arnoux, 21-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
elorvc	|- ( ( ph /\ z e. dom X ) -> ( z e. ( X∘RV/𝑐 R A ) <-> ( X ` z ) R A ) )

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

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

Proof of Theorem elorvc

Step	Hyp	Ref	Expression
1		orvcval.1	. . . . 5 |- ( ph -> Fun X )
2		orvcval.2	. . . . 5 |- ( ph -> X e. V )
3		orvcval.3	. . . . 5 |- ( ph -> A e. W )
4	1, 2, 3	orvcval2 30520	. . . 4 |- ( ph -> ( X∘RV/𝑐 R A ) = { z e. dom X | ( X ` z ) R A } )
5	4	eleq2d 2687	. . 3 |- ( ph -> ( z e. ( X∘RV/𝑐 R A ) <-> z e. { z e. dom X | ( X ` z ) R A } ) )
6		rabid 3116	. . 3 |- ( z e. { z e. dom X | ( X ` z ) R A } <-> ( z e. dom X /\ ( X ` z ) R A ) )
7	5, 6	syl6bb 276	. 2 |- ( ph -> ( z e. ( X∘RV/𝑐 R A ) <-> ( z e. dom X /\ ( X ` z ) R A ) ) )
8	7	baibd 948	1 |- ( ( ph /\ z e. dom X ) -> ( z e. ( X∘RV/𝑐 R A ) <-> ( X ` z ) R A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   {crab 2916   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   ` cfv 5888  (class class class)co 6650  ∘_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-ne 2795  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-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-orvc 30518

This theorem is referenced by:  elorrvc  30525