Theorem ofmpteq 6916

Description: Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)

Assertion

Ref	Expression
ofmpteq	|- ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) -> ( ( x e. A |-> B ) oF R ( x e. A |-> C ) ) = ( x e. A |-> ( B R C ) ) )

Distinct variable groups:   x, A   x, R

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem ofmpteq

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) -> A e. V )
2		simpr 477	. . . 4 |- ( ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) /\ a e. A ) -> a e. A )
3		simpl2 1065	. . . . 5 |- ( ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) /\ a e. A ) -> ( x e. A |-> B ) Fn A )
4		eqid 2622	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
5	4	mptfng 6019	. . . . 5 |- ( A. x e. A B e. _V <-> ( x e. A |-> B ) Fn A )
6	3, 5	sylibr 224	. . . 4 |- ( ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) /\ a e. A ) -> A. x e. A B e. _V )
7		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ a / x ]_ B
8	7	nfel1 2779	. . . . 5 |- F/ x [_ a / x ]_ B e. _V
9		csbeq1a 3542	. . . . . 6 |- ( x = a -> B = [_ a / x ]_ B )
10	9	eleq1d 2686	. . . . 5 |- ( x = a -> ( B e. _V <-> [_ a / x ]_ B e. _V ) )
11	8, 10	rspc 3303	. . . 4 |- ( a e. A -> ( A. x e. A B e. _V -> [_ a / x ]_ B e. _V ) )
12	2, 6, 11	sylc 65	. . 3 |- ( ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) /\ a e. A ) -> [_ a / x ]_ B e. _V )
13		simpl3 1066	. . . . 5 |- ( ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) /\ a e. A ) -> ( x e. A |-> C ) Fn A )
14		eqid 2622	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
15	14	mptfng 6019	. . . . 5 |- ( A. x e. A C e. _V <-> ( x e. A |-> C ) Fn A )
16	13, 15	sylibr 224	. . . 4 |- ( ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) /\ a e. A ) -> A. x e. A C e. _V )
17		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ a / x ]_ C
18	17	nfel1 2779	. . . . 5 |- F/ x [_ a / x ]_ C e. _V
19		csbeq1a 3542	. . . . . 6 |- ( x = a -> C = [_ a / x ]_ C )
20	19	eleq1d 2686	. . . . 5 |- ( x = a -> ( C e. _V <-> [_ a / x ]_ C e. _V ) )
21	18, 20	rspc 3303	. . . 4 |- ( a e. A -> ( A. x e. A C e. _V -> [_ a / x ]_ C e. _V ) )
22	2, 16, 21	sylc 65	. . 3 |- ( ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) /\ a e. A ) -> [_ a / x ]_ C e. _V )
23		nfcv 2764	. . . . 5 |- F/_ a B
24	23, 7, 9	cbvmpt 4749	. . . 4 |- ( x e. A |-> B ) = ( a e. A |-> [_ a / x ]_ B )
25	24	a1i 11	. . 3 |- ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) -> ( x e. A |-> B ) = ( a e. A |-> [_ a / x ]_ B ) )
26		nfcv 2764	. . . . 5 |- F/_ a C
27	26, 17, 19	cbvmpt 4749	. . . 4 |- ( x e. A |-> C ) = ( a e. A |-> [_ a / x ]_ C )
28	27	a1i 11	. . 3 |- ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) -> ( x e. A |-> C ) = ( a e. A |-> [_ a / x ]_ C ) )
29	1, 12, 22, 25, 28	offval2 6914	. 2 |- ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) -> ( ( x e. A |-> B ) oF R ( x e. A |-> C ) ) = ( a e. A |-> ( [_ a / x ]_ B R [_ a / x ]_ C ) ) )
30		nfcv 2764	. . 3 |- F/_ a ( B R C )
31		nfcv 2764	. . . 4 |- F/_ x R
32	7, 31, 17	nfov 6676	. . 3 |- F/_ x ( [_ a / x ]_ B R [_ a / x ]_ C )
33	9, 19	oveq12d 6668	. . 3 |- ( x = a -> ( B R C ) = ( [_ a / x ]_ B R [_ a / x ]_ C ) )
34	30, 32, 33	cbvmpt 4749	. 2 |- ( x e. A |-> ( B R C ) ) = ( a e. A |-> ( [_ a / x ]_ B R [_ a / x ]_ C ) )
35	29, 34	syl6eqr 2674	1 |- ( ( A e. V /\ ( x e. A |-> B ) Fn A /\ ( x e. A |-> C ) Fn A ) -> ( ( x e. A |-> B ) oF R ( x e. A |-> C ) ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   |-> cmpt 4729   Fn wfn 5883  (class class class)co 6650   oFcof 6895

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

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-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-of 6897

This theorem is referenced by:  mdetrlin  20408  mzpaddmpt  37304  mzpmulmpt  37305  mzpcompact2lem  37314