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Theorem fnofval 5644
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 (φA 𝑉)
offval.4 (φB 𝑊)
offval.5 (AB) = 𝑆
ofval.6 ((φ 𝑋 A) → (𝐹𝑋) = 𝐶)
ofval.7 ((φ 𝑋 B) → (𝐺𝑋) = 𝐷)
ofval.8 (φ𝑅 Fn (𝑈 × 𝑉))
ofval.9 (φ𝐶 𝑈)
ofval.10 (φ𝐷 𝑉)
Assertion
Ref Expression
fnofval ((φ 𝑋 𝑆) → ((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 (φA 𝑉)
4 offval.4 . . . . 5 (φB 𝑊)
5 offval.5 . . . . 5 (AB) = 𝑆
6 eqidd 2023 . . . . 5 ((φ x A) → (𝐹x) = (𝐹x))
7 eqidd 2023 . . . . 5 ((φ x B) → (𝐺x) = (𝐺x))
81, 2, 3, 4, 5, 6, 7offval 5642 . . . 4 (φ → (𝐹𝑓 𝑅𝐺) = (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))))
98fveq1d 5105 . . 3 (φ → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))‘𝑋))
109adantr 261 . 2 ((φ 𝑋 𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))‘𝑋))
11 ax-ia2 100 . . 3 ((φ 𝑋 𝑆) → 𝑋 𝑆)
12 ofval.8 . . . . 5 (φ𝑅 Fn (𝑈 × 𝑉))
1312adantr 261 . . . 4 ((φ 𝑋 𝑆) → 𝑅 Fn (𝑈 × 𝑉))
14 ofval.9 . . . . . 6 (φ𝐶 𝑈)
1514adantr 261 . . . . 5 ((φ 𝑋 𝑆) → 𝐶 𝑈)
16 inss1 3134 . . . . . . . . 9 (AB) ⊆ A
175, 16eqsstr3i 2953 . . . . . . . 8 𝑆A
1817sseli 2918 . . . . . . 7 (𝑋 𝑆𝑋 A)
19 ofval.6 . . . . . . 7 ((φ 𝑋 A) → (𝐹𝑋) = 𝐶)
2018, 19sylan2 270 . . . . . 6 ((φ 𝑋 𝑆) → (𝐹𝑋) = 𝐶)
2120eleq1d 2088 . . . . 5 ((φ 𝑋 𝑆) → ((𝐹𝑋) 𝑈𝐶 𝑈))
2215, 21mpbird 156 . . . 4 ((φ 𝑋 𝑆) → (𝐹𝑋) 𝑈)
23 ofval.10 . . . . . 6 (φ𝐷 𝑉)
2423adantr 261 . . . . 5 ((φ 𝑋 𝑆) → 𝐷 𝑉)
25 inss2 3135 . . . . . . . . 9 (AB) ⊆ B
265, 25eqsstr3i 2953 . . . . . . . 8 𝑆B
2726sseli 2918 . . . . . . 7 (𝑋 𝑆𝑋 B)
28 ofval.7 . . . . . . 7 ((φ 𝑋 B) → (𝐺𝑋) = 𝐷)
2927, 28sylan2 270 . . . . . 6 ((φ 𝑋 𝑆) → (𝐺𝑋) = 𝐷)
3029eleq1d 2088 . . . . 5 ((φ 𝑋 𝑆) → ((𝐺𝑋) 𝑉𝐷 𝑉))
3124, 30mpbird 156 . . . 4 ((φ 𝑋 𝑆) → (𝐺𝑋) 𝑉)
32 fnovex 5462 . . . 4 ((𝑅 Fn (𝑈 × 𝑉) (𝐹𝑋) 𝑈 (𝐺𝑋) 𝑉) → ((𝐹𝑋)𝑅(𝐺𝑋)) V)
3313, 22, 31, 32syl3anc 1121 . . 3 ((φ 𝑋 𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) V)
34 fveq2 5103 . . . . 5 (x = 𝑋 → (𝐹x) = (𝐹𝑋))
35 fveq2 5103 . . . . 5 (x = 𝑋 → (𝐺x) = (𝐺𝑋))
3634, 35oveq12d 5454 . . . 4 (x = 𝑋 → ((𝐹x)𝑅(𝐺x)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
37 eqid 2022 . . . 4 (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) = (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))
3836, 37fvmptg 5173 . . 3 ((𝑋 𝑆 ((𝐹𝑋)𝑅(𝐺𝑋)) V) → ((x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
3911, 33, 38syl2anc 393 . 2 ((φ 𝑋 𝑆) → ((x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
4020, 29oveq12d 5454 . 2 ((φ 𝑋 𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
4110, 39, 403eqtrd 2058 1 ((φ 𝑋 𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  Vcvv 2535  cin 2893  cmpt 3792   × cxp 4270   Fn wfn 4824  cfv 4829  (class class class)co 5436  𝑓 cof 5633
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-of 5635
This theorem is referenced by: (None)
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