Theorem offval2 5726

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 2392	. . . . 5 |- ( ph -> A. x e. A B e. W )
3		eqid 2040	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5025	. . . . 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 4989	. . . 4 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 156	. . 3 |- ( ph -> F Fn A )
9		offval2.3	. . . . . 6 |- ( ( ph /\ x e. A ) -> C e. X )
10	9	ralrimiva 2392	. . . . 5 |- ( ph -> A. x e. A C e. X )
11		eqid 2040	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
12	11	fnmpt 5025	. . . . 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 4989	. . . 4 |- ( ph -> ( G Fn A <-> ( x e. A |-> C ) Fn A ) )
16	13, 15	mpbird 156	. . 3 |- ( ph -> G Fn A )
17		offval2.1	. . 3 |- ( ph -> A e. V )
18		inidm 3146	. . 3 |- ( A i^i A ) = A
19	6	adantr 261	. . . 4 |- ( ( ph /\ y e. A ) -> F = ( x e. A |-> B ) )
20	19	fveq1d 5180	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( ( x e. A |-> B ) ` y ) )
21	14	adantr 261	. . . 4 |- ( ( ph /\ y e. A ) -> G = ( x e. A |-> C ) )
22	21	fveq1d 5180	. . 3 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( ( x e. A |-> C ) ` y ) )
23	8, 16, 17, 17, 18, 20, 22	offval 5719	. 2 |- ( ph -> ( F oF R G ) = ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) )
24		nffvmpt1 5186	. . . . 5 |- F/_ x ( ( x e. A |-> B ) ` y )
25		nfcv 2178	. . . . 5 |- F/_ x R
26		nffvmpt1 5186	. . . . 5 |- F/_ x ( ( x e. A |-> C ) ` y )
27	24, 25, 26	nfov 5535	. . . 4 |- F/_ x ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) )
28		nfcv 2178	. . . 4 |- F/_ y ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) )
29		fveq2 5178	. . . . 5 |- ( y = x -> ( ( x e. A |-> B ) ` y ) = ( ( x e. A |-> B ) ` x ) )
30		fveq2 5178	. . . . 5 |- ( y = x -> ( ( x e. A |-> C ) ` y ) = ( ( x e. A |-> C ) ` x ) )
31	29, 30	oveq12d 5530	. . . 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 3851	. . 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 103	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. A )
34	3	fvmpt2 5254	. . . . . 6 |- ( ( x e. A /\ B e. W ) -> ( ( x e. A |-> B ) ` x ) = B )
35	33, 1, 34	syl2anc 391	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
36	11	fvmpt2 5254	. . . . . 6 |- ( ( x e. A /\ C e. X ) -> ( ( x e. A |-> C ) ` x ) = C )
37	33, 9, 36	syl2anc 391	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> C ) ` x ) = C )
38	35, 37	oveq12d 5530	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) = ( B R C ) )
39	38	mpteq2dva 3847	. . 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	syl5eq 2084	. 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 2072	1 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   |-> cmpt 3818   Fn wfn 4897   ` cfv 4902  (class class class)co 5512   oFcof 5710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-of 5712

This theorem is referenced by:  ofc12  5731  caofinvl  5733  caofcom  5734