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Theorem offval 5661
Description: Value of an operation applied to two functions. (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
offval.6 ((φ x A) → (𝐹x) = 𝐶)
offval.7 ((φ x B) → (𝐺x) = 𝐷)
Assertion
Ref Expression
offval (φ → (𝐹𝑓 𝑅𝐺) = (x 𝑆 ↦ (𝐶𝑅𝐷)))
Distinct variable groups:   x,A   x,𝐹   x,𝐺   φ,x   x,𝑆   x,𝑅
Allowed substitution hints:   B(x)   𝐶(x)   𝐷(x)   𝑉(x)   𝑊(x)

Proof of Theorem offval
Dummy variables f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4 (φ𝐹 Fn A)
2 offval.3 . . . 4 (φA 𝑉)
3 fnex 5326 . . . 4 ((𝐹 Fn A A 𝑉) → 𝐹 V)
41, 2, 3syl2anc 391 . . 3 (φ𝐹 V)
5 offval.2 . . . 4 (φ𝐺 Fn B)
6 offval.4 . . . 4 (φB 𝑊)
7 fnex 5326 . . . 4 ((𝐺 Fn B B 𝑊) → 𝐺 V)
85, 6, 7syl2anc 391 . . 3 (φ𝐺 V)
9 fndm 4941 . . . . . . . 8 (𝐹 Fn A → dom 𝐹 = A)
101, 9syl 14 . . . . . . 7 (φ → dom 𝐹 = A)
11 fndm 4941 . . . . . . . 8 (𝐺 Fn B → dom 𝐺 = B)
125, 11syl 14 . . . . . . 7 (φ → dom 𝐺 = B)
1310, 12ineq12d 3133 . . . . . 6 (φ → (dom 𝐹 ∩ dom 𝐺) = (AB))
14 offval.5 . . . . . 6 (AB) = 𝑆
1513, 14syl6eq 2085 . . . . 5 (φ → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
1615mpteq1d 3833 . . . 4 (φ → (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))) = (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))))
17 inex1g 3884 . . . . . 6 (A 𝑉 → (AB) V)
1814, 17syl5eqelr 2122 . . . . 5 (A 𝑉𝑆 V)
19 mptexg 5329 . . . . 5 (𝑆 V → (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) V)
202, 18, 193syl 17 . . . 4 (φ → (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) V)
2116, 20eqeltrd 2111 . . 3 (φ → (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))) V)
22 dmeq 4478 . . . . . 6 (f = 𝐹 → dom f = dom 𝐹)
23 dmeq 4478 . . . . . 6 (g = 𝐺 → dom g = dom 𝐺)
2422, 23ineqan12d 3134 . . . . 5 ((f = 𝐹 g = 𝐺) → (dom f ∩ dom g) = (dom 𝐹 ∩ dom 𝐺))
25 fveq1 5120 . . . . . 6 (f = 𝐹 → (fx) = (𝐹x))
26 fveq1 5120 . . . . . 6 (g = 𝐺 → (gx) = (𝐺x))
2725, 26oveqan12d 5474 . . . . 5 ((f = 𝐹 g = 𝐺) → ((fx)𝑅(gx)) = ((𝐹x)𝑅(𝐺x)))
2824, 27mpteq12dv 3830 . . . 4 ((f = 𝐹 g = 𝐺) → (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))) = (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))))
29 df-of 5654 . . . 4 𝑓 𝑅 = (f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
3028, 29ovmpt2ga 5572 . . 3 ((𝐹 V 𝐺 V (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))) V) → (𝐹𝑓 𝑅𝐺) = (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))))
314, 8, 21, 30syl3anc 1134 . 2 (φ → (𝐹𝑓 𝑅𝐺) = (x (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹x)𝑅(𝐺x))))
3214eleq2i 2101 . . . . 5 (x (AB) ↔ x 𝑆)
33 elin 3120 . . . . 5 (x (AB) ↔ (x A x B))
3432, 33bitr3i 175 . . . 4 (x 𝑆 ↔ (x A x B))
35 offval.6 . . . . . 6 ((φ x A) → (𝐹x) = 𝐶)
3635adantrr 448 . . . . 5 ((φ (x A x B)) → (𝐹x) = 𝐶)
37 offval.7 . . . . . 6 ((φ x B) → (𝐺x) = 𝐷)
3837adantrl 447 . . . . 5 ((φ (x A x B)) → (𝐺x) = 𝐷)
3936, 38oveq12d 5473 . . . 4 ((φ (x A x B)) → ((𝐹x)𝑅(𝐺x)) = (𝐶𝑅𝐷))
4034, 39sylan2b 271 . . 3 ((φ x 𝑆) → ((𝐹x)𝑅(𝐺x)) = (𝐶𝑅𝐷))
4140mpteq2dva 3838 . 2 (φ → (x 𝑆 ↦ ((𝐹x)𝑅(𝐺x))) = (x 𝑆 ↦ (𝐶𝑅𝐷)))
4231, 16, 413eqtrd 2073 1 (φ → (𝐹𝑓 𝑅𝐺) = (x 𝑆 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  cmpt 3809  dom cdm 4288   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-bnd 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:  fnofval  5663  off  5666  ofres  5667  offval2  5668  suppssof1  5670  ofco  5671  offveqb  5672
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