Theorem ofvalg 6168

Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)

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 ∧ 𝑋 ∈ 𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)

Assertion

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

Proof of Theorem ofvalg

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 2206	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2206	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 6166	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9	8	fveq1d 5578	. . 3 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
10	9	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
11		eqid 2205	. . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
12		fveq2 5576	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
13		fveq2 5576	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
14	12, 13	oveq12d 5962	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
15		simpr 110	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
16		inss1 3393	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
17	5, 16	eqsstrri 3226	. . . . . . 7 ⊢ 𝑆 ⊆ 𝐴
18	17	sseli 3189	. . . . . 6 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴)
19		ofval.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
20	18, 19	sylan2 286	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
21		inss2 3394	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
22	5, 21	eqsstrri 3226	. . . . . . 7 ⊢ 𝑆 ⊆ 𝐵
23	22	sseli 3189	. . . . . 6 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵)
24		ofval.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
25	23, 24	sylan2 286	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
26	20, 25	oveq12d 5962	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷))
27		ofval.8	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)
28	26, 27	eqeltrd 2282	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ 𝑈)
29	11, 14, 15, 28	fvmptd3 5673	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
30	10, 29, 26	3eqtrd 2242	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176   ∩ cin 3165   ↦ cmpt 4105   Fn wfn 5266  ‘cfv 5271  (class class class)co 5944   ∘_𝑓 cof 6156

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 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585

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 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-of 6158

This theorem is referenced by:  offeq  6172  ofc1g  6180  ofc2g  6181  ofnegsub  9035  gsumfzmptfidmadd  13675  mplsubgfilemcl  14461  dvaddxxbr  15173  dvmulxxbr  15174  plyaddlem1  15219