Theorem offval2 6246

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

Hypotheses

Ref	Expression
offval2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval2.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
offval2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
offval2.4	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
offval2.5	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))

Assertion

Ref	Expression
offval2	⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Distinct variable groups:   𝑥,𝐴   𝜑,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 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
2	1	ralrimiva 2603	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
3		eqid 2229	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	fnmpt 5456	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	syl 14	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		offval2.4	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
7	6	fneq1d 5417	. . . 4 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴))
8	5, 7	mpbird 167	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9		offval2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
10	9	ralrimiva 2603	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑋)
11		eqid 2229	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
12	11	fnmpt 5456	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
13	10, 12	syl 14	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
14		offval2.5	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	14	fneq1d 5417	. . . 4 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
16	13, 15	mpbird 167	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
17		offval2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
18		inidm 3414	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
19	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
20	19	fveq1d 5637	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
21	14	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))
22	21	fveq1d 5637	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
23	8, 16, 17, 17, 18, 20, 22	offval 6238	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑦 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))))
24		nffvmpt1 5646	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
25		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑥𝑅
26		nffvmpt1 5646	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
27	24, 25, 26	nfov 6043	. . . 4 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
28		nfcv 2372	. . . 4 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
29		fveq2 5635	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
30		fveq2 5635	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
31	29, 30	oveq12d 6031	. . . 4 ⊢ (𝑦 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)) = (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)))
32	27, 28, 31	cbvmpt 4182	. . 3 ⊢ (𝑦 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))) = (𝑥 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)))
33		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
34	3	fvmpt2 5726	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
35	33, 1, 34	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
36	11	fvmpt2 5726	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
37	33, 9, 36	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
38	35, 37	oveq12d 6031	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)) = (𝐵𝑅𝐶))
39	38	mpteq2dva 4177	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
40	32, 39	eqtrid 2274	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
41	23, 40	eqtrd 2262	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ↦ cmpt 4148   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013   ∘_𝑓 cof 6228

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230

This theorem is referenced by:  ofc12  6254  caofinvl  6256  caofcom  6261  caofdig  6264  pwsplusgval  13368  pwsmulrval  13369  pwssub  13686  gsumfzmptfidmadd  13916  gsumfzmptfidmadd2  13917  psrlinv  14688  dvimulf  15420  dvexp  15425  dvmptaddx  15433  dvmptmulx  15434  dvef  15441  plyaddlem1  15461  plymullem1  15462  plycolemc  15472  lgseisenlem3  15791  lgseisenlem4  15792