Theorem msubfval 31421

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 31420	. . . . 5 |- ( T e. _V -> S = ( f e. ( R ^pm V ) |-> ( e e. E |-> <. ( 1st ` e ) , ( ( O ` f ) ` ( 2nd ` e ) ) >. ) ) )
7	6	adantr 481	. . . 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 792	. . . . . . . 8 |- ( ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) /\ e e. E ) -> f = F )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) /\ e e. E ) -> ( O ` f ) = ( O ` F ) )
10	9	fveq1d 6193	. . . . . 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 4409	. . . . 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 4744	. . . 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 6201	. . . . . . . 8 |- (mREx ` T ) e. _V
14	2, 13	eqeltri 2697	. . . . . . 7 |- R e. _V
15		fvex 6201	. . . . . . . 8 |- (mVR ` T ) e. _V
16	1, 15	eqeltri 2697	. . . . . . 7 |- V e. _V
17	14, 16	pm3.2i 471	. . . . . 6 |- ( R e. _V /\ V e. _V )
18	17	a1i 11	. . . . 5 |- ( T e. _V -> ( R e. _V /\ V e. _V ) )
19		elpm2r 7875	. . . . 5 |- ( ( ( R e. _V /\ V e. _V ) /\ ( F : A --> R /\ A C_ V ) ) -> F e. ( R ^pm V ) )
20	18, 19	sylan 488	. . . 4 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> F e. ( R ^pm V ) )
21		fvex 6201	. . . . . . 7 |- (mEx ` T ) e. _V
22	4, 21	eqeltri 2697	. . . . . 6 |- E e. _V
23	22	mptex 6486	. . . . 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 6288	. . 3 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )
26	25	ex 450	. 2 |- ( T e. _V -> ( ( F : A --> R /\ A C_ V ) -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) ) )
27		0fv 6227	. . . . 5 |- ( (/) ` F ) = (/)
28		mpt0 6021	. . . . 5 |- ( e e. (/) |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) = (/)
29	27, 28	eqtr4i 2647	. . . 4 |- ( (/) ` F ) = ( e e. (/) |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. )
30		fvprc 6185	. . . . . 6 |- ( -. T e. _V -> (mSubst ` T ) = (/) )
31	3, 30	syl5eq 2668	. . . . 5 |- ( -. T e. _V -> S = (/) )
32	31	fveq1d 6193	. . . 4 |- ( -. T e. _V -> ( S ` F ) = ( (/) ` F ) )
33		fvprc 6185	. . . . . 6 |- ( -. T e. _V -> (mEx ` T ) = (/) )
34	4, 33	syl5eq 2668	. . . . 5 |- ( -. T e. _V -> E = (/) )
35	34	mpteq1d 4738	. . . 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 2682	. . 3 |- ( -. T e. _V -> ( S ` F ) = ( e e. E |-> <. ( 1st ` e ) , ( ( O ` F ) ` ( 2nd ` e ) ) >. ) )
37	36	a1d 25	. 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 176	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 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^pm cpm 7858  mVRcmvar 31358  mRExcmrex 31363  mExcmex 31364  mRSubstcmrsub 31367  mSubstcmsub 31368

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-rep 4771  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-reu 2919  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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-msub 31388

This theorem is referenced by:  msubval  31422  msubrn  31426