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Theorem fnofval 5798
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 2084 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2084 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 5796 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 5253 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 270 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 simpr 108 . . 3 ((𝜑𝑋𝑆) → 𝑋𝑆)
12 ofval.8 . . . . 5 (𝜑𝑅 Fn (𝑈 × 𝑉))
1312adantr 270 . . . 4 ((𝜑𝑋𝑆) → 𝑅 Fn (𝑈 × 𝑉))
14 ofval.9 . . . . . 6 (𝜑𝐶𝑈)
1514adantr 270 . . . . 5 ((𝜑𝑋𝑆) → 𝐶𝑈)
16 inss1 3204 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
175, 16eqsstr3i 3041 . . . . . . . 8 𝑆𝐴
1817sseli 3006 . . . . . . 7 (𝑋𝑆𝑋𝐴)
19 ofval.6 . . . . . . 7 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2018, 19sylan2 280 . . . . . 6 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
2120eleq1d 2151 . . . . 5 ((𝜑𝑋𝑆) → ((𝐹𝑋) ∈ 𝑈𝐶𝑈))
2215, 21mpbird 165 . . . 4 ((𝜑𝑋𝑆) → (𝐹𝑋) ∈ 𝑈)
23 ofval.10 . . . . . 6 (𝜑𝐷𝑉)
2423adantr 270 . . . . 5 ((𝜑𝑋𝑆) → 𝐷𝑉)
25 inss2 3205 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐵
265, 25eqsstr3i 3041 . . . . . . . 8 𝑆𝐵
2726sseli 3006 . . . . . . 7 (𝑋𝑆𝑋𝐵)
28 ofval.7 . . . . . . 7 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2927, 28sylan2 280 . . . . . 6 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
3029eleq1d 2151 . . . . 5 ((𝜑𝑋𝑆) → ((𝐺𝑋) ∈ 𝑉𝐷𝑉))
3124, 30mpbird 165 . . . 4 ((𝜑𝑋𝑆) → (𝐺𝑋) ∈ 𝑉)
32 fnovex 5615 . . . 4 ((𝑅 Fn (𝑈 × 𝑉) ∧ (𝐹𝑋) ∈ 𝑈 ∧ (𝐺𝑋) ∈ 𝑉) → ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V)
3313, 22, 31, 32syl3anc 1170 . . 3 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V)
34 fveq2 5251 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
35 fveq2 5251 . . . . 5 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
3634, 35oveq12d 5607 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
37 eqid 2083 . . . 4 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
3836, 37fvmptg 5323 . . 3 ((𝑋𝑆 ∧ ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
3911, 33, 38syl2anc 403 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
4020, 29oveq12d 5607 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
4110, 39, 403eqtrd 2119 1 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  Vcvv 2612  cin 2983  cmpt 3865   × cxp 4397   Fn wfn 4962  cfv 4967  (class class class)co 5589  𝑓 cof 5787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-setind 4315
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-of 5789
This theorem is referenced by: (None)
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