Theorem orvcval4 30522

Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 30519. (Contributed by Thierry Arnoux, 21-Jan-2017.)

Hypotheses

Ref	Expression
orvccel.1	|- ( ph -> S e. U. ran sigAlgebra )
orvccel.2	|- ( ph -> J e. Top )
orvccel.3	|- ( ph -> X e. ( SMblFnM (sigaGen ` J ) ) )
orvccel.4	|- ( ph -> A e. V )

Assertion

Ref	Expression
orvcval4	|- ( ph -> ( X∘RV/𝑐 R A ) = ( `' X " { y e. U. J | y R A } ) )

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

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

Proof of Theorem orvcval4

Step	Hyp	Ref	Expression
1		orvccel.1	. . . . 5 |- ( ph -> S e. U. ran sigAlgebra )
2		orvccel.2	. . . . . 6 |- ( ph -> J e. Top )
3	2	sgsiga 30205	. . . . 5 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
4		orvccel.3	. . . . 5 |- ( ph -> X e. ( SMblFnM (sigaGen ` J ) ) )
5	1, 3, 4	isanmbfm 30318	. . . 4 |- ( ph -> X e. U. ran MblFnM )
6	5	mbfmfun 30316	. . 3 |- ( ph -> Fun X )
7	1, 3, 4	mbfmf 30317	. . . . 5 |- ( ph -> X : U. S --> U. (sigaGen ` J ) )
8		elex 3212	. . . . . . 7 |- ( J e. Top -> J e. _V )
9		unisg 30206	. . . . . . 7 |- ( J e. _V -> U. (sigaGen ` J ) = U. J )
10	2, 8, 9	3syl 18	. . . . . 6 |- ( ph -> U. (sigaGen ` J ) = U. J )
11	10	feq3d 6032	. . . . 5 |- ( ph -> ( X : U. S --> U. (sigaGen ` J ) <-> X : U. S --> U. J ) )
12	7, 11	mpbid 222	. . . 4 |- ( ph -> X : U. S --> U. J )
13		frn 6053	. . . 4 |- ( X : U. S --> U. J -> ran X C_ U. J )
14	12, 13	syl 17	. . 3 |- ( ph -> ran X C_ U. J )
15		fimacnvinrn2 6349	. . 3 |- ( ( Fun X /\ ran X C_ U. J ) -> ( `' X " { y | y R A } ) = ( `' X " ( { y | y R A } i^i U. J ) ) )
16	6, 14, 15	syl2anc 693	. 2 |- ( ph -> ( `' X " { y | y R A } ) = ( `' X " ( { y | y R A } i^i U. J ) ) )
17		orvccel.4	. . 3 |- ( ph -> A e. V )
18	6, 4, 17	orvcval 30519	. 2 |- ( ph -> ( X∘RV/𝑐 R A ) = ( `' X " { y | y R A } ) )
19		dfrab2 3903	. . . 4 |- { y e. U. J | y R A } = ( { y | y R A } i^i U. J )
20	19	a1i 11	. . 3 |- ( ph -> { y e. U. J | y R A } = ( { y | y R A } i^i U. J ) )
21	20	imaeq2d 5466	. 2 |- ( ph -> ( `' X " { y e. U. J | y R A } ) = ( `' X " ( { y | y R A } i^i U. J ) ) )
22	16, 18, 21	3eqtr4d 2666	1 |- ( ph -> ( X∘RV/𝑐 R A ) = ( `' X " { y e. U. J | y R A } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698  sigAlgebracsiga 30170  sigaGencsigagen 30201  MblFnMcmbfm 30312  ∘_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-fal 1489  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-csb 3534  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-int 4476  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-siga 30171  df-sigagen 30202  df-mbfm 30313  df-orvc 30518

This theorem is referenced by:  orvcoel  30523  orvccel  30524  orrvcval4  30526