ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ov Structured version   GIF version

Theorem ov 5562
Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ov.1 𝐶 V
ov.2 (x = A → (φψ))
ov.3 (y = B → (ψχ))
ov.4 (z = 𝐶 → (χθ))
ov.5 ((x 𝑅 y 𝑆) → ∃!zφ)
ov.6 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}
Assertion
Ref Expression
ov ((A 𝑅 B 𝑆) → ((A𝐹B) = 𝐶θ))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   x,𝑅,y,z   x,𝑆,y,z   θ,x,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)   χ(x,y,z)   𝐹(x,y,z)

Proof of Theorem ov
StepHypRef Expression
1 df-ov 5458 . . . . 5 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 ov.6 . . . . . 6 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}
32fveq1i 5122 . . . . 5 (𝐹‘⟨A, B⟩) = ({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩)
41, 3eqtri 2057 . . . 4 (A𝐹B) = ({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩)
54eqeq1i 2044 . . 3 ((A𝐹B) = 𝐶 ↔ ({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶)
6 ov.5 . . . . . 6 ((x 𝑅 y 𝑆) → ∃!zφ)
76fnoprab 5546 . . . . 5 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)}
8 eleq1 2097 . . . . . . . 8 (x = A → (x 𝑅A 𝑅))
98anbi1d 438 . . . . . . 7 (x = A → ((x 𝑅 y 𝑆) ↔ (A 𝑅 y 𝑆)))
10 eleq1 2097 . . . . . . . 8 (y = B → (y 𝑆B 𝑆))
1110anbi2d 437 . . . . . . 7 (y = B → ((A 𝑅 y 𝑆) ↔ (A 𝑅 B 𝑆)))
129, 11opelopabg 3996 . . . . . 6 ((A 𝑅 B 𝑆) → (⟨A, B {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)} ↔ (A 𝑅 B 𝑆)))
1312ibir 166 . . . . 5 ((A 𝑅 B 𝑆) → ⟨A, B {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)})
14 fnopfvb 5158 . . . . 5 (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)} A, B {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)}) → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}))
157, 13, 14sylancr 393 . . . 4 ((A 𝑅 B 𝑆) → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}))
16 ov.1 . . . . 5 𝐶 V
17 ov.2 . . . . . . 7 (x = A → (φψ))
189, 17anbi12d 442 . . . . . 6 (x = A → (((x 𝑅 y 𝑆) φ) ↔ ((A 𝑅 y 𝑆) ψ)))
19 ov.3 . . . . . . 7 (y = B → (ψχ))
2011, 19anbi12d 442 . . . . . 6 (y = B → (((A 𝑅 y 𝑆) ψ) ↔ ((A 𝑅 B 𝑆) χ)))
21 ov.4 . . . . . . 7 (z = 𝐶 → (χθ))
2221anbi2d 437 . . . . . 6 (z = 𝐶 → (((A 𝑅 B 𝑆) χ) ↔ ((A 𝑅 B 𝑆) θ)))
2318, 20, 22eloprabg 5534 . . . . 5 ((A 𝑅 B 𝑆 𝐶 V) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} ↔ ((A 𝑅 B 𝑆) θ)))
2416, 23mp3an3 1220 . . . 4 ((A 𝑅 B 𝑆) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} ↔ ((A 𝑅 B 𝑆) θ)))
2515, 24bitrd 177 . . 3 ((A 𝑅 B 𝑆) → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ((A 𝑅 B 𝑆) θ)))
265, 25syl5bb 181 . 2 ((A 𝑅 B 𝑆) → ((A𝐹B) = 𝐶 ↔ ((A 𝑅 B 𝑆) θ)))
2726bianabs 543 1 ((A 𝑅 B 𝑆) → ((A𝐹B) = 𝐶θ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  ∃!weu 1897  Vcvv 2551  cop 3370  {copab 3808   Fn wfn 4840  cfv 4845  (class class class)co 5455  {coprab 5456
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-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-v 2553  df-sbc 2759  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-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458  df-oprab 5459
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator