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Theorem fnofval 5663
 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 2038 . . . . 5 ((φ x A) → (𝐹x) = (𝐹x))
7 eqidd 2038 . . . . 5 ((φ x B) → (𝐺x) = (𝐺x))
81, 2, 3, 4, 5, 6, 7offval 5661 . . . 4 (φ → (𝐹𝑓 𝑅𝐺) = (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))))
98fveq1d 5123 . . 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 simpr 103 . . 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 3151 . . . . . . . . 9 (AB) ⊆ A
175, 16eqsstr3i 2970 . . . . . . . 8 𝑆A
1817sseli 2935 . . . . . . 7 (𝑋 𝑆𝑋 A)
19 ofval.6 . . . . . . 7 ((φ 𝑋 A) → (𝐹𝑋) = 𝐶)
2018, 19sylan2 270 . . . . . 6 ((φ 𝑋 𝑆) → (𝐹𝑋) = 𝐶)
2120eleq1d 2103 . . . . 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 3152 . . . . . . . . 9 (AB) ⊆ B
265, 25eqsstr3i 2970 . . . . . . . 8 𝑆B
2726sseli 2935 . . . . . . 7 (𝑋 𝑆𝑋 B)
28 ofval.7 . . . . . . 7 ((φ 𝑋 B) → (𝐺𝑋) = 𝐷)
2927, 28sylan2 270 . . . . . 6 ((φ 𝑋 𝑆) → (𝐺𝑋) = 𝐷)
3029eleq1d 2103 . . . . 5 ((φ 𝑋 𝑆) → ((𝐺𝑋) 𝑉𝐷 𝑉))
3124, 30mpbird 156 . . . 4 ((φ 𝑋 𝑆) → (𝐺𝑋) 𝑉)
32 fnovex 5481 . . . 4 ((𝑅 Fn (𝑈 × 𝑉) (𝐹𝑋) 𝑈 (𝐺𝑋) 𝑉) → ((𝐹𝑋)𝑅(𝐺𝑋)) V)
3313, 22, 31, 32syl3anc 1134 . . 3 ((φ 𝑋 𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) V)
34 fveq2 5121 . . . . 5 (x = 𝑋 → (𝐹x) = (𝐹𝑋))
35 fveq2 5121 . . . . 5 (x = 𝑋 → (𝐺x) = (𝐺𝑋))
3634, 35oveq12d 5473 . . . 4 (x = 𝑋 → ((𝐹x)𝑅(𝐺x)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
37 eqid 2037 . . . 4 (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) = (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))
3836, 37fvmptg 5191 . . 3 ((𝑋 𝑆 ((𝐹𝑋)𝑅(𝐺𝑋)) V) → ((x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
3911, 33, 38syl2anc 391 . 2 ((φ 𝑋 𝑆) → ((x 𝑆 ↦ ((𝐹x)𝑅(𝐺x)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
4020, 29oveq12d 5473 . 2 ((φ 𝑋 𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
4110, 39, 403eqtrd 2073 1 ((φ 𝑋 𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910   ↦ cmpt 3809   × cxp 4286   Fn wfn 4840  ‘cfv 4845  (class class class)co 5455   ∘𝑓 cof 5652 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654 This theorem is referenced by: (None)
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