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Theorem ovg 5581
Description: The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
ovg.1 (x = A → (φψ))
ovg.2 (y = B → (ψχ))
ovg.3 (z = 𝐶 → (χθ))
ovg.4 ((τ (x 𝑅 y 𝑆)) → ∃!zφ)
ovg.5 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}
Assertion
Ref Expression
ovg ((τ (A 𝑅 B 𝑆 𝐶 𝐷)) → ((A𝐹B) = 𝐶θ))
Distinct variable groups:   ψ,x   χ,x,y   θ,x,y,z   τ,x,y   x,𝑅,y,z   x,𝑆,y,z   x,A,y,z   x,B,y,z   x,𝐶,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(y,z)   χ(z)   τ(z)   𝐷(x,y,z)   𝐹(x,y,z)

Proof of Theorem ovg
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-ov 5458 . . . . 5 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 ovg.5 . . . . . 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 eqeq2 2046 . . . . . . . . . 10 (𝑐 = 𝐶 → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝑐 ↔ ({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶))
7 opeq2 3541 . . . . . . . . . . 11 (𝑐 = 𝐶 → ⟨⟨A, B⟩, 𝑐⟩ = ⟨⟨A, B⟩, 𝐶⟩)
87eleq1d 2103 . . . . . . . . . 10 (𝑐 = 𝐶 → (⟨⟨A, B⟩, 𝑐 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}))
96, 8bibi12d 224 . . . . . . . . 9 (𝑐 = 𝐶 → ((({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝑐 ↔ ⟨⟨A, B⟩, 𝑐 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}) ↔ (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)})))
109imbi2d 219 . . . . . . . 8 (𝑐 = 𝐶 → (((τ (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)})) ↔ ((τ (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)}))))
11 ovg.4 . . . . . . . . . . . 12 ((τ (x 𝑅 y 𝑆)) → ∃!zφ)
1211ex 108 . . . . . . . . . . 11 (τ → ((x 𝑅 y 𝑆) → ∃!zφ))
1312alrimivv 1752 . . . . . . . . . 10 (τxy((x 𝑅 y 𝑆) → ∃!zφ))
14 fnoprabg 5544 . . . . . . . . . 10 (xy((x 𝑅 y 𝑆) → ∃!zφ) → {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)})
1513, 14syl 14 . . . . . . . . 9 (τ → {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)})
16 eleq1 2097 . . . . . . . . . . . 12 (x = A → (x 𝑅A 𝑅))
1716anbi1d 438 . . . . . . . . . . 11 (x = A → ((x 𝑅 y 𝑆) ↔ (A 𝑅 y 𝑆)))
18 eleq1 2097 . . . . . . . . . . . 12 (y = B → (y 𝑆B 𝑆))
1918anbi2d 437 . . . . . . . . . . 11 (y = B → ((A 𝑅 y 𝑆) ↔ (A 𝑅 B 𝑆)))
2017, 19opelopabg 3996 . . . . . . . . . 10 ((A 𝑅 B 𝑆) → (⟨A, B {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)} ↔ (A 𝑅 B 𝑆)))
2120ibir 166 . . . . . . . . 9 ((A 𝑅 B 𝑆) → ⟨A, B {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)})
22 fnopfvb 5158 . . . . . . . . 9 (({⟨⟨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)}))
2315, 21, 22syl2an 273 . . . . . . . 8 ((τ (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)}))
2410, 23vtoclg 2607 . . . . . . 7 (𝐶 𝐷 → ((τ (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)})))
2524com12 27 . . . . . 6 ((τ (A 𝑅 B 𝑆)) → (𝐶 𝐷 → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)})))
2625exp32 347 . . . . 5 (τ → (A 𝑅 → (B 𝑆 → (𝐶 𝐷 → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)})))))
27263imp2 1118 . . . 4 ((τ (A 𝑅 B 𝑆 𝐶 𝐷)) → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}))
28 ovg.1 . . . . . . 7 (x = A → (φψ))
2917, 28anbi12d 442 . . . . . 6 (x = A → (((x 𝑅 y 𝑆) φ) ↔ ((A 𝑅 y 𝑆) ψ)))
30 ovg.2 . . . . . . 7 (y = B → (ψχ))
3119, 30anbi12d 442 . . . . . 6 (y = B → (((A 𝑅 y 𝑆) ψ) ↔ ((A 𝑅 B 𝑆) χ)))
32 ovg.3 . . . . . . 7 (z = 𝐶 → (χθ))
3332anbi2d 437 . . . . . 6 (z = 𝐶 → (((A 𝑅 B 𝑆) χ) ↔ ((A 𝑅 B 𝑆) θ)))
3429, 31, 33eloprabg 5534 . . . . 5 ((A 𝑅 B 𝑆 𝐶 𝐷) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} ↔ ((A 𝑅 B 𝑆) θ)))
3534adantl 262 . . . 4 ((τ (A 𝑅 B 𝑆 𝐶 𝐷)) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} ↔ ((A 𝑅 B 𝑆) θ)))
3627, 35bitrd 177 . . 3 ((τ (A 𝑅 B 𝑆 𝐶 𝐷)) → (({⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}‘⟨A, B⟩) = 𝐶 ↔ ((A 𝑅 B 𝑆) θ)))
375, 36syl5bb 181 . 2 ((τ (A 𝑅 B 𝑆 𝐶 𝐷)) → ((A𝐹B) = 𝐶 ↔ ((A 𝑅 B 𝑆) θ)))
38 biidd 161 . . . . 5 ((A 𝑅 B 𝑆) → (((A 𝑅 B 𝑆) θ) ↔ ((A 𝑅 B 𝑆) θ)))
3938bianabs 543 . . . 4 ((A 𝑅 B 𝑆) → (((A 𝑅 B 𝑆) θ) ↔ θ))
40393adant3 923 . . 3 ((A 𝑅 B 𝑆 𝐶 𝐷) → (((A 𝑅 B 𝑆) θ) ↔ θ))
4140adantl 262 . 2 ((τ (A 𝑅 B 𝑆 𝐶 𝐷)) → (((A 𝑅 B 𝑆) θ) ↔ θ))
4237, 41bitrd 177 1 ((τ (A 𝑅 B 𝑆 𝐶 𝐷)) → ((A𝐹B) = 𝐶θ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884  wal 1240   = wceq 1242   wcel 1390  ∃!weu 1897  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)
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