Theorem msubfval 30212

Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
msubffval.v	|- V = (mVR ` T )
msubffval.r	|- R = (mREx ` T )
msubffval.s	|- S = (mSubst ` T )
msubffval.e	|- E = (mEx ` T )
msubffval.o	|- O = (mRSubst ` T )

Assertion

Ref	Expression
msubfval	|- ( ( F : A --> R /\ A C_ V ) -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )

Distinct variable groups:   e, E   e, O   R, e   T, e   e, V   A, e   e, F

Allowed substitution hint:   S( e)

Proof of Theorem msubfval

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		msubffval.v	. . . . . 6 |- V = (mVR ` T )
2		msubffval.r	. . . . . 6 |- R = (mREx ` T )
3		msubffval.s	. . . . . 6 |- S = (mSubst ` T )
4		msubffval.e	. . . . . 6 |- E = (mEx ` T )
5		msubffval.o	. . . . . 6 |- O = (mRSubst ` T )
6	1, 2, 3, 4, 5	msubffval 30211	. . . . 5 |- ( T e. _V -> S = ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( O ` f ) ` ( 2nd ` e ) ) >. ) ) )
7	6	adantr 471	. . . 4 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> S = ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( O ` f ) ` ( 2nd ` e ) ) >. ) ) )
8		simplr 767	. . . . . . . 8 |- ( ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) /\ e e. E ) -> f = F )
9	8	fveq2d 5896	. . . . . . 7 |- ( ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) /\ e e. E ) -> ( O ` f ) = ( O ` F ) )
10	9	fveq1d 5894	. . . . . 6 |- ( ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) /\ e e. E ) -> ( ( O ` f ) ` ( 2nd ` e ) ) = ( ( O ` F ) ` ( 2nd ` e ) ) )
11	10	opeq2d 4187	. . . . 5 |- ( ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) /\ e e. E ) -> <. ( 1st ` e ) , ( ( O ` f ) ` ( 2nd ` e ) ) >. = <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. )
12	11	mpteq2dva 4505	. . . 4 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> ( e e. E |-> <. ( 1st ` e ) , ( ( O ` f ) ` ( 2nd ` e ) ) >. ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )
13		fvex 5902	. . . . . . . 8 |- (mREx ` T ) e. _V
14	2, 13	eqeltri 2536	. . . . . . 7 |- R e. _V
15		fvex 5902	. . . . . . . 8 |- (mVR ` T ) e. _V
16	1, 15	eqeltri 2536	. . . . . . 7 |- V e. _V
17	14, 16	pm3.2i 461	. . . . . 6 |- ( R e. _V /\ V e. _V )
18	17	a1i 11	. . . . 5 |- ( T e. _V -> ( R e. _V /\ V e. _V ) )
19		elpm2r 7520	. . . . 5 |- ( ( ( R e. _V /\ V e. _V ) /\ ( F : A --> R /\ A C_ V ) ) -> F e. ( R ^pm V ) )
20	18, 19	sylan 478	. . . 4 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> F e. ( R ^pm V ) )
21		fvex 5902	. . . . . . 7 |- (mEx ` T ) e. _V
22	4, 21	eqeltri 2536	. . . . . 6 |- E e. _V
23	22	mptex 6166	. . . . 5 |- ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) e. _V
24	23	a1i 11	. . . 4 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) e. _V )
25	7, 12, 20, 24	fvmptd 5982	. . 3 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )
26	25	ex 440	. 2 |- ( T e. _V -> ( ( F : A --> R /\ A C_ V ) -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) ) )
27		0fv 5925	. . . . 5 |- ( (/) ` F ) = (/)
28		mpt0 5731	. . . . 5 |- ( e e. (/) |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) = (/)
29	27, 28	eqtr4i 2487	. . . 4 |- ( (/) ` F ) = ( e e. (/) |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. )
30		fvprc 5886	. . . . . 6 |- ( -. T e. _V -> (mSubst ` T ) = (/) )
31	3, 30	syl5eq 2508	. . . . 5 |- ( -. T e. _V -> S = (/) )
32	31	fveq1d 5894	. . . 4 |- ( -. T e. _V -> ( S ` F ) = ( (/) ` F ) )
33		fvprc 5886	. . . . . 6 |- ( -. T e. _V -> (mEx ` T ) = (/) )
34	4, 33	syl5eq 2508	. . . . 5 |- ( -. T e. _V -> E = (/) )
35	34	mpteq1d 4500	. . . 4 |- ( -. T e. _V -> ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) = ( e e. (/) |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )
36	29, 32, 35	3eqtr4a 2522	. . 3 |- ( -. T e. _V -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )
37	36	a1d 26	. 2 |- ( -. T e. _V -> ( ( F : A --> R /\ A C_ V ) -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) ) )
38	26, 37	pm2.61i 169	1 |- ( ( F : A --> R /\ A C_ V ) -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   = wceq 1455   e. wcel 1898   _Vcvv 3057   C_ wss 3416   (/)c0 3743   <.cop 3986   |-> cmpt 4477   -->wf 5601   ` cfv 5605  (class class class)co 6320   1stc1st 6823   2ndc2nd 6824   ^pm cpm 7504  mVRcmvar 30149  mRExcmrex 30154  mExcmex 30155  mRSubstcmrsub 30158  mSubstcmsub 30159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-pm 7506  df-msub 30179

This theorem is referenced by:  msubval  30213  msubrn  30217