Theorem fnofval 5741

Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
offval.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
offval.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)
offval.5	⊢ (𝐴 ∩ 𝐵) = 𝑆
ofval.6	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
ofval.7	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
ofval.8	⊢ (𝜑 → 𝑅 Fn (𝑈 × 𝑉))
ofval.9	⊢ (𝜑 → 𝐶 ∈ 𝑈)
ofval.10	⊢ (𝜑 → 𝐷 ∈ 𝑉)

Assertion

Ref	Expression
fnofval	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Proof of Theorem fnofval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		offval.2	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		offval.3	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		offval.4	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		offval.5	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = 𝑆
6		eqidd 2082	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2082	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 5739	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9	8	fveq1d 5200	. . 3 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
10	9	adantr 270	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
11		simpr 108	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
12		ofval.8	. . . . 5 ⊢ (𝜑 → 𝑅 Fn (𝑈 × 𝑉))
13	12	adantr 270	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝑅 Fn (𝑈 × 𝑉))
14		ofval.9	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
15	14	adantr 270	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝐶 ∈ 𝑈)
16		inss1 3186	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
17	5, 16	eqsstr3i 3030	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝐴
18	17	sseli 2995	. . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴)
19		ofval.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
20	18, 19	sylan2 280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
21	20	eleq1d 2147	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋) ∈ 𝑈 ↔ 𝐶 ∈ 𝑈))
22	15, 21	mpbird 165	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝑈)
23		ofval.10	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
24	23	adantr 270	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝐷 ∈ 𝑉)
25		inss2 3187	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
26	5, 25	eqsstr3i 3030	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝐵
27	26	sseli 2995	. . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵)
28		ofval.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
29	27, 28	sylan2 280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
30	29	eleq1d 2147	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐺‘𝑋) ∈ 𝑉 ↔ 𝐷 ∈ 𝑉))
31	24, 30	mpbird 165	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) ∈ 𝑉)
32		fnovex 5558	. . . 4 ⊢ ((𝑅 Fn (𝑈 × 𝑉) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ (𝐺‘𝑋) ∈ 𝑉) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V)
33	13, 22, 31, 32	syl3anc 1169	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V)
34		fveq2 5198	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
35		fveq2 5198	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
36	34, 35	oveq12d 5550	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
37		eqid 2081	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
38	36, 37	fvmptg 5269	. . 3 ⊢ ((𝑋 ∈ 𝑆 ∧ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
39	11, 33, 38	syl2anc 403	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
40	20, 29	oveq12d 5550	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷))
41	10, 39, 40	3eqtrd 2117	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∩ cin 2972   ↦ cmpt 3839   × cxp 4361   Fn wfn 4917  ‘cfv 4922  (class class class)co 5532   ∘_𝑓 cof 5730

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-of 5732

This theorem is referenced by: (None)