Theorem offval2 6251

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 2605	. . . . 5 |- ( ph -> A. x e. A B e. W )
3		eqid 2231	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5459	. . . . 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 5420	. . . 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 2605	. . . . 5 |- ( ph -> A. x e. A C e. X )
11		eqid 2231	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
12	11	fnmpt 5459	. . . . 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 5420	. . . 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 3416	. . 3 |- ( A i^i A ) = A
19	6	adantr 276	. . . 4 |- ( ( ph /\ y e. A ) -> F = ( x e. A |-> B ) )
20	19	fveq1d 5641	. . 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 5641	. . 3 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( ( x e. A |-> C ) ` y ) )
23	8, 16, 17, 17, 18, 20, 22	offval 6243	. 2 |- ( ph -> ( F oF R G ) = ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) )
24		nffvmpt1 5650	. . . . 5 |- F/_ x ( ( x e. A |-> B ) ` y )
25		nfcv 2374	. . . . 5 |- F/_ x R
26		nffvmpt1 5650	. . . . 5 |- F/_ x ( ( x e. A |-> C ) ` y )
27	24, 25, 26	nfov 6048	. . . 4 |- F/_ x ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) )
28		nfcv 2374	. . . 4 |- F/_ y ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) )
29		fveq2 5639	. . . . 5 |- ( y = x -> ( ( x e. A |-> B ) ` y ) = ( ( x e. A |-> B ) ` x ) )
30		fveq2 5639	. . . . 5 |- ( y = x -> ( ( x e. A |-> C ) ` y ) = ( ( x e. A |-> C ) ` x ) )
31	29, 30	oveq12d 6036	. . . 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 4184	. . 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 5730	. . . . . 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 5730	. . . . . 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 6036	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) = ( B R C ) )
39	38	mpteq2dva 4179	. . 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 2276	. 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 2264	1 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   |-> cmpt 4150   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   oFcof 6233

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235

This theorem is referenced by:  ofc12  6259  caofinvl  6261  caofcom  6266  caofdig  6269  pwsplusgval  13396  pwsmulrval  13397  pwssub  13714  gsumfzmptfidmadd  13944  gsumfzmptfidmadd2  13945  psrlinv  14717  dvimulf  15449  dvexp  15454  dvmptaddx  15462  dvmptmulx  15463  dvef  15470  plyaddlem1  15490  plymullem1  15491  plycolemc  15501  lgseisenlem3  15820  lgseisenlem4  15821