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Theorem ofrfval 5662
Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-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
ofrfval (φ → (𝐹𝑟 𝑅𝐺x 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 ofrfval
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 dmeq 4478 . . . . . 6 (f = 𝐹 → dom f = dom 𝐹)
10 dmeq 4478 . . . . . 6 (g = 𝐺 → dom g = dom 𝐺)
119, 10ineqan12d 3134 . . . . 5 ((f = 𝐹 g = 𝐺) → (dom f ∩ dom g) = (dom 𝐹 ∩ dom 𝐺))
12 fveq1 5120 . . . . . 6 (f = 𝐹 → (fx) = (𝐹x))
13 fveq1 5120 . . . . . 6 (g = 𝐺 → (gx) = (𝐺x))
1412, 13breqan12d 3770 . . . . 5 ((f = 𝐹 g = 𝐺) → ((fx)𝑅(gx) ↔ (𝐹x)𝑅(𝐺x)))
1511, 14raleqbidv 2511 . . . 4 ((f = 𝐹 g = 𝐺) → (x (dom f ∩ dom g)(fx)𝑅(gx) ↔ x (dom 𝐹 ∩ dom 𝐺)(𝐹x)𝑅(𝐺x)))
16 df-ofr 5655 . . . 4 𝑟 𝑅 = {⟨f, g⟩ ∣ x (dom f ∩ dom g)(fx)𝑅(gx)}
1715, 16brabga 3992 . . 3 ((𝐹 V 𝐺 V) → (𝐹𝑟 𝑅𝐺x (dom 𝐹 ∩ dom 𝐺)(𝐹x)𝑅(𝐺x)))
184, 8, 17syl2anc 391 . 2 (φ → (𝐹𝑟 𝑅𝐺x (dom 𝐹 ∩ dom 𝐺)(𝐹x)𝑅(𝐺x)))
19 fndm 4941 . . . . . 6 (𝐹 Fn A → dom 𝐹 = A)
201, 19syl 14 . . . . 5 (φ → dom 𝐹 = A)
21 fndm 4941 . . . . . 6 (𝐺 Fn B → dom 𝐺 = B)
225, 21syl 14 . . . . 5 (φ → dom 𝐺 = B)
2320, 22ineq12d 3133 . . . 4 (φ → (dom 𝐹 ∩ dom 𝐺) = (AB))
24 offval.5 . . . 4 (AB) = 𝑆
2523, 24syl6eq 2085 . . 3 (φ → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2625raleqdv 2505 . 2 (φ → (x (dom 𝐹 ∩ dom 𝐺)(𝐹x)𝑅(𝐺x) ↔ x 𝑆 (𝐹x)𝑅(𝐺x)))
27 inss1 3151 . . . . . . 7 (AB) ⊆ A
2824, 27eqsstr3i 2970 . . . . . 6 𝑆A
2928sseli 2935 . . . . 5 (x 𝑆x A)
30 offval.6 . . . . 5 ((φ x A) → (𝐹x) = 𝐶)
3129, 30sylan2 270 . . . 4 ((φ x 𝑆) → (𝐹x) = 𝐶)
32 inss2 3152 . . . . . . 7 (AB) ⊆ B
3324, 32eqsstr3i 2970 . . . . . 6 𝑆B
3433sseli 2935 . . . . 5 (x 𝑆x B)
35 offval.7 . . . . 5 ((φ x B) → (𝐺x) = 𝐷)
3634, 35sylan2 270 . . . 4 ((φ x 𝑆) → (𝐺x) = 𝐷)
3731, 36breq12d 3768 . . 3 ((φ x 𝑆) → ((𝐹x)𝑅(𝐺x) ↔ 𝐶𝑅𝐷))
3837ralbidva 2316 . 2 (φ → (x 𝑆 (𝐹x)𝑅(𝐺x) ↔ x 𝑆 𝐶𝑅𝐷))
3918, 26, 383bitrd 203 1 (φ → (𝐹𝑟 𝑅𝐺x 𝑆 𝐶𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  cin 2910   class class class wbr 3755  dom cdm 4288   Fn wfn 4840  cfv 4845  𝑟 cofr 5653
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-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
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  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-ofr 5655
This theorem is referenced by:  ofrval  5664  ofrfval2  5669  caofref  5674  caofrss  5677  caoftrn  5678
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