Theorem offval2 6097

Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
offval2.1	|- ( ph -> A e. V )
offval2.2	|- ( ( ph /\ x e. A ) -> B e. W )
offval2.3	|- ( ( ph /\ x e. A ) -> C e. X )
offval2.4	|- ( ph -> F = ( x e. A |-> B ) )
offval2.5	|- ( ph -> G = ( x e. A |-> C ) )

Assertion

Ref	Expression
offval2	|- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Distinct variable groups:   x, A   ph, x   x, R

Allowed substitution hints:   B( x)   C( x)   F( x)   G( x)   V( x)   W( x)   X( x)

Proof of Theorem offval2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval2.2	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. W )
2	1	ralrimiva 2550	. . . . 5 |- ( ph -> A. x e. A B e. W )
3		eqid 2177	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5342	. . . . 5 |- ( A. x e. A B e. W -> ( x e. A |-> B ) Fn A )
5	2, 4	syl 14	. . . 4 |- ( ph -> ( x e. A |-> B ) Fn A )
6		offval2.4	. . . . 5 |- ( ph -> F = ( x e. A |-> B ) )
7	6	fneq1d 5306	. . . 4 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 167	. . 3 |- ( ph -> F Fn A )
9		offval2.3	. . . . . 6 |- ( ( ph /\ x e. A ) -> C e. X )
10	9	ralrimiva 2550	. . . . 5 |- ( ph -> A. x e. A C e. X )
11		eqid 2177	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
12	11	fnmpt 5342	. . . . 5 |- ( A. x e. A C e. X -> ( x e. A |-> C ) Fn A )
13	10, 12	syl 14	. . . 4 |- ( ph -> ( x e. A |-> C ) Fn A )
14		offval2.5	. . . . 5 |- ( ph -> G = ( x e. A |-> C ) )
15	14	fneq1d 5306	. . . 4 |- ( ph -> ( G Fn A <-> ( x e. A |-> C ) Fn A ) )
16	13, 15	mpbird 167	. . 3 |- ( ph -> G Fn A )
17		offval2.1	. . 3 |- ( ph -> A e. V )
18		inidm 3344	. . 3 |- ( A i^i A ) = A
19	6	adantr 276	. . . 4 |- ( ( ph /\ y e. A ) -> F = ( x e. A |-> B ) )
20	19	fveq1d 5517	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( ( x e. A |-> B ) ` y ) )
21	14	adantr 276	. . . 4 |- ( ( ph /\ y e. A ) -> G = ( x e. A |-> C ) )
22	21	fveq1d 5517	. . 3 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( ( x e. A |-> C ) ` y ) )
23	8, 16, 17, 17, 18, 20, 22	offval 6089	. 2 |- ( ph -> ( F oF R G ) = ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) )
24		nffvmpt1 5526	. . . . 5 |- F/_ x ( ( x e. A |-> B ) ` y )
25		nfcv 2319	. . . . 5 |- F/_ x R
26		nffvmpt1 5526	. . . . 5 |- F/_ x ( ( x e. A |-> C ) ` y )
27	24, 25, 26	nfov 5904	. . . 4 |- F/_ x ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) )
28		nfcv 2319	. . . 4 |- F/_ y ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) )
29		fveq2 5515	. . . . 5 |- ( y = x -> ( ( x e. A |-> B ) ` y ) = ( ( x e. A |-> B ) ` x ) )
30		fveq2 5515	. . . . 5 |- ( y = x -> ( ( x e. A |-> C ) ` y ) = ( ( x e. A |-> C ) ` x ) )
31	29, 30	oveq12d 5892	. . . 4 |- ( y = x -> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) = ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) )
32	27, 28, 31	cbvmpt 4098	. . 3 |- ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) = ( x e. A |-> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) )
33		simpr 110	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. A )
34	3	fvmpt2 5599	. . . . . 6 |- ( ( x e. A /\ B e. W ) -> ( ( x e. A |-> B ) ` x ) = B )
35	33, 1, 34	syl2anc 411	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
36	11	fvmpt2 5599	. . . . . 6 |- ( ( x e. A /\ C e. X ) -> ( ( x e. A |-> C ) ` x ) = C )
37	33, 9, 36	syl2anc 411	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> C ) ` x ) = C )
38	35, 37	oveq12d 5892	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) = ( B R C ) )
39	38	mpteq2dva 4093	. . 3 |- ( ph -> ( x e. A |-> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) ) = ( x e. A |-> ( B R C ) ) )
40	32, 39	eqtrid 2222	. 2 |- ( ph -> ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) = ( x e. A |-> ( B R C ) ) )
41	23, 40	eqtrd 2210	1 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   |-> cmpt 4064   Fn wfn 5211   ` cfv 5216  (class class class)co 5874   oFcof 6080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-of 6082

This theorem is referenced by:  ofc12  6102  caofinvl  6104  caofcom  6105  dvimulf  14063  dvexp  14068  dvmptaddx  14074  dvmptmulx  14075  dvef  14081