Theorem offval2 6176

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 2579	. . . . 5 |- ( ph -> A. x e. A B e. W )
3		eqid 2205	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5404	. . . . 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 5365	. . . 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 2579	. . . . 5 |- ( ph -> A. x e. A C e. X )
11		eqid 2205	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
12	11	fnmpt 5404	. . . . 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 5365	. . . 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 3382	. . 3 |- ( A i^i A ) = A
19	6	adantr 276	. . . 4 |- ( ( ph /\ y e. A ) -> F = ( x e. A |-> B ) )
20	19	fveq1d 5580	. . 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 5580	. . 3 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( ( x e. A |-> C ) ` y ) )
23	8, 16, 17, 17, 18, 20, 22	offval 6168	. 2 |- ( ph -> ( F oF R G ) = ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) )
24		nffvmpt1 5589	. . . . 5 |- F/_ x ( ( x e. A |-> B ) ` y )
25		nfcv 2348	. . . . 5 |- F/_ x R
26		nffvmpt1 5589	. . . . 5 |- F/_ x ( ( x e. A |-> C ) ` y )
27	24, 25, 26	nfov 5976	. . . 4 |- F/_ x ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) )
28		nfcv 2348	. . . 4 |- F/_ y ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) )
29		fveq2 5578	. . . . 5 |- ( y = x -> ( ( x e. A |-> B ) ` y ) = ( ( x e. A |-> B ) ` x ) )
30		fveq2 5578	. . . . 5 |- ( y = x -> ( ( x e. A |-> C ) ` y ) = ( ( x e. A |-> C ) ` x ) )
31	29, 30	oveq12d 5964	. . . 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 4140	. . 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 5665	. . . . . 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 5665	. . . . . 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 5964	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) = ( B R C ) )
39	38	mpteq2dva 4135	. . 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 2250	. 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 2238	1 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   |-> cmpt 4106   Fn wfn 5267   ` cfv 5272  (class class class)co 5946   oFcof 6158

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-of 6160

This theorem is referenced by:  ofc12  6184  caofinvl  6186  caofcom  6191  caofdig  6194  pwsplusgval  13160  pwsmulrval  13161  pwssub  13478  gsumfzmptfidmadd  13708  gsumfzmptfidmadd2  13709  psrlinv  14479  dvimulf  15211  dvexp  15216  dvmptaddx  15224  dvmptmulx  15225  dvef  15232  plyaddlem1  15252  plymullem1  15253  plycolemc  15263  lgseisenlem3  15582  lgseisenlem4  15583