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Theorem ofvalg 6092
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 2178 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2178 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 6090 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 5518 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 276 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 eqid 2177 . . 3 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
12 fveq2 5516 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
13 fveq2 5516 . . . 4 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1412, 13oveq12d 5893 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
15 simpr 110 . . 3 ((𝜑𝑋𝑆) → 𝑋𝑆)
16 inss1 3356 . . . . . . . 8 (𝐴𝐵) ⊆ 𝐴
175, 16eqsstrri 3189 . . . . . . 7 𝑆𝐴
1817sseli 3152 . . . . . 6 (𝑋𝑆𝑋𝐴)
19 ofval.6 . . . . . 6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2018, 19sylan2 286 . . . . 5 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
21 inss2 3357 . . . . . . . 8 (𝐴𝐵) ⊆ 𝐵
225, 21eqsstrri 3189 . . . . . . 7 𝑆𝐵
2322sseli 3152 . . . . . 6 (𝑋𝑆𝑋𝐵)
24 ofval.7 . . . . . 6 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2523, 24sylan2 286 . . . . 5 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
2620, 25oveq12d 5893 . . . 4 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
27 ofval.8 . . . 4 ((𝜑𝑋𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)
2826, 27eqeltrd 2254 . . 3 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ 𝑈)
2911, 14, 15, 28fvmptd3 5610 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
3010, 29, 263eqtrd 2214 1 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  cin 3129  cmpt 4065   Fn wfn 5212  cfv 5217  (class class class)co 5875  𝑓 cof 6081
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-of 6083
This theorem is referenced by:  offeq  6096  dvaddxxbr  14168  dvmulxxbr  14169
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