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Theorem fnofval 5721
 Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
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 (𝜑𝑅 Fn (𝑈 × 𝑉))
ofval.9 (𝜑𝐶𝑈)
ofval.10 (𝜑𝐷𝑉)
Assertion
Ref Expression
fnofval ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Proof of Theorem fnofval
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 2041 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2041 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 5719 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 5180 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 261 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 simpr 103 . . 3 ((𝜑𝑋𝑆) → 𝑋𝑆)
12 ofval.8 . . . . 5 (𝜑𝑅 Fn (𝑈 × 𝑉))
1312adantr 261 . . . 4 ((𝜑𝑋𝑆) → 𝑅 Fn (𝑈 × 𝑉))
14 ofval.9 . . . . . 6 (𝜑𝐶𝑈)
1514adantr 261 . . . . 5 ((𝜑𝑋𝑆) → 𝐶𝑈)
16 inss1 3157 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
175, 16eqsstr3i 2976 . . . . . . . 8 𝑆𝐴
1817sseli 2941 . . . . . . 7 (𝑋𝑆𝑋𝐴)
19 ofval.6 . . . . . . 7 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2018, 19sylan2 270 . . . . . 6 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
2120eleq1d 2106 . . . . 5 ((𝜑𝑋𝑆) → ((𝐹𝑋) ∈ 𝑈𝐶𝑈))
2215, 21mpbird 156 . . . 4 ((𝜑𝑋𝑆) → (𝐹𝑋) ∈ 𝑈)
23 ofval.10 . . . . . 6 (𝜑𝐷𝑉)
2423adantr 261 . . . . 5 ((𝜑𝑋𝑆) → 𝐷𝑉)
25 inss2 3158 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐵
265, 25eqsstr3i 2976 . . . . . . . 8 𝑆𝐵
2726sseli 2941 . . . . . . 7 (𝑋𝑆𝑋𝐵)
28 ofval.7 . . . . . . 7 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2927, 28sylan2 270 . . . . . 6 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
3029eleq1d 2106 . . . . 5 ((𝜑𝑋𝑆) → ((𝐺𝑋) ∈ 𝑉𝐷𝑉))
3124, 30mpbird 156 . . . 4 ((𝜑𝑋𝑆) → (𝐺𝑋) ∈ 𝑉)
32 fnovex 5538 . . . 4 ((𝑅 Fn (𝑈 × 𝑉) ∧ (𝐹𝑋) ∈ 𝑈 ∧ (𝐺𝑋) ∈ 𝑉) → ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V)
3313, 22, 31, 32syl3anc 1135 . . 3 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V)
34 fveq2 5178 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
35 fveq2 5178 . . . . 5 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
3634, 35oveq12d 5530 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
37 eqid 2040 . . . 4 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
3836, 37fvmptg 5248 . . 3 ((𝑋𝑆 ∧ ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
3911, 33, 38syl2anc 391 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
4020, 29oveq12d 5530 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
4110, 39, 403eqtrd 2076 1 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∩ cin 2916   ↦ cmpt 3818   × cxp 4343   Fn wfn 4897  ‘cfv 4902  (class class class)co 5512   ∘𝑓 cof 5710 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-of 5712 This theorem is referenced by: (None)
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