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Theorem ofvalg 6035
 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 (𝜑𝐹 Fn 𝐴)
offval.2 (𝜑𝐺 Fn 𝐵)
offval.3 (𝜑𝐴𝑉)
offval.4 (𝜑𝐵𝑊)
offval.5 (𝐴𝐵) = 𝑆
ofval.6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
ofval.7 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
ofval.8 ((𝜑𝑋𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)
Assertion
Ref Expression
ofvalg ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Proof of Theorem ofvalg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . 5 (𝜑𝐹 Fn 𝐴)
2 offval.2 . . . . 5 (𝜑𝐺 Fn 𝐵)
3 offval.3 . . . . 5 (𝜑𝐴𝑉)
4 offval.4 . . . . 5 (𝜑𝐵𝑊)
5 offval.5 . . . . 5 (𝐴𝐵) = 𝑆
6 eqidd 2158 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2158 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 6033 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 5467 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 274 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 eqid 2157 . . 3 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
12 fveq2 5465 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
13 fveq2 5465 . . . 4 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1412, 13oveq12d 5836 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
15 simpr 109 . . 3 ((𝜑𝑋𝑆) → 𝑋𝑆)
16 inss1 3327 . . . . . . . 8 (𝐴𝐵) ⊆ 𝐴
175, 16eqsstrri 3161 . . . . . . 7 𝑆𝐴
1817sseli 3124 . . . . . 6 (𝑋𝑆𝑋𝐴)
19 ofval.6 . . . . . 6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2018, 19sylan2 284 . . . . 5 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
21 inss2 3328 . . . . . . . 8 (𝐴𝐵) ⊆ 𝐵
225, 21eqsstrri 3161 . . . . . . 7 𝑆𝐵
2322sseli 3124 . . . . . 6 (𝑋𝑆𝑋𝐵)
24 ofval.7 . . . . . 6 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2523, 24sylan2 284 . . . . 5 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
2620, 25oveq12d 5836 . . . 4 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
27 ofval.8 . . . 4 ((𝜑𝑋𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)
2826, 27eqeltrd 2234 . . 3 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ 𝑈)
2911, 14, 15, 28fvmptd3 5558 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
3010, 29, 263eqtrd 2194 1 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128   ∩ cin 3101   ↦ cmpt 4025   Fn wfn 5162  ‘cfv 5167  (class class class)co 5818   ∘𝑓 cof 6024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-setind 4494 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-of 6026 This theorem is referenced by:  offeq  6039  dvaddxxbr  13025  dvmulxxbr  13026
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