Theorem fnofval 5663

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 2038	. . . . 5 ⊢ ((φ ∧ x ∈ A) → (𝐹‘x) = (𝐹‘x))
7		eqidd 2038	. . . . 5 ⊢ ((φ ∧ x ∈ B) → (𝐺‘x) = (𝐺‘x))
8	1, 2, 3, 4, 5, 6, 7	offval 5661	. . . 4 ⊢ (φ → (𝐹 ∘𝑓 𝑅𝐺) = (x ∈ 𝑆 ↦ ((𝐹‘x)𝑅(𝐺‘x))))
9	8	fveq1d 5123	. . 3 ⊢ (φ → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((x ∈ 𝑆 ↦ ((𝐹‘x)𝑅(𝐺‘x)))‘𝑋))
10	9	adantr 261	. 2 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((x ∈ 𝑆 ↦ ((𝐹‘x)𝑅(𝐺‘x)))‘𝑋))
11		simpr 103	. . 3 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
12		ofval.8	. . . . 5 ⊢ (φ → 𝑅 Fn (𝑈 × 𝑉))
13	12	adantr 261	. . . 4 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → 𝑅 Fn (𝑈 × 𝑉))
14		ofval.9	. . . . . 6 ⊢ (φ → 𝐶 ∈ 𝑈)
15	14	adantr 261	. . . . 5 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → 𝐶 ∈ 𝑈)
16		inss1 3151	. . . . . . . . 9 ⊢ (A ∩ B) ⊆ A
17	5, 16	eqsstr3i 2970	. . . . . . . 8 ⊢ 𝑆 ⊆ A
18	17	sseli 2935	. . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ A)
19		ofval.6	. . . . . . 7 ⊢ ((φ ∧ 𝑋 ∈ A) → (𝐹‘𝑋) = 𝐶)
20	18, 19	sylan2 270	. . . . . 6 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
21	20	eleq1d 2103	. . . . 5 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋) ∈ 𝑈 ↔ 𝐶 ∈ 𝑈))
22	15, 21	mpbird 156	. . . 4 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝑈)
23		ofval.10	. . . . . 6 ⊢ (φ → 𝐷 ∈ 𝑉)
24	23	adantr 261	. . . . 5 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → 𝐷 ∈ 𝑉)
25		inss2 3152	. . . . . . . . 9 ⊢ (A ∩ B) ⊆ B
26	5, 25	eqsstr3i 2970	. . . . . . . 8 ⊢ 𝑆 ⊆ B
27	26	sseli 2935	. . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ B)
28		ofval.7	. . . . . . 7 ⊢ ((φ ∧ 𝑋 ∈ B) → (𝐺‘𝑋) = 𝐷)
29	27, 28	sylan2 270	. . . . . 6 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
30	29	eleq1d 2103	. . . . 5 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → ((𝐺‘𝑋) ∈ 𝑉 ↔ 𝐷 ∈ 𝑉))
31	24, 30	mpbird 156	. . . 4 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) ∈ 𝑉)
32		fnovex 5481	. . . 4 ⊢ ((𝑅 Fn (𝑈 × 𝑉) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ (𝐺‘𝑋) ∈ 𝑉) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V)
33	13, 22, 31, 32	syl3anc 1134	. . 3 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V)
34		fveq2 5121	. . . . 5 ⊢ (x = 𝑋 → (𝐹‘x) = (𝐹‘𝑋))
35		fveq2 5121	. . . . 5 ⊢ (x = 𝑋 → (𝐺‘x) = (𝐺‘𝑋))
36	34, 35	oveq12d 5473	. . . 4 ⊢ (x = 𝑋 → ((𝐹‘x)𝑅(𝐺‘x)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
37		eqid 2037	. . . 4 ⊢ (x ∈ 𝑆 ↦ ((𝐹‘x)𝑅(𝐺‘x))) = (x ∈ 𝑆 ↦ ((𝐹‘x)𝑅(𝐺‘x)))
38	36, 37	fvmptg 5191	. . 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 391	. 2 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → ((x ∈ 𝑆 ↦ ((𝐹‘x)𝑅(𝐺‘x)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
40	20, 29	oveq12d 5473	. 2 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷))
41	10, 39, 40	3eqtrd 2073	1 ⊢ ((φ ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910   ↦ cmpt 3809   × cxp 4286   Fn wfn 4840  ‘cfv 4845  (class class class)co 5455   ∘_𝑓 cof 5652

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654

This theorem is referenced by: (None)