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Theorem ovid 5559
Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovid.1 ((x 𝑅 y 𝑆) → ∃!zφ)
ovid.2 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}
Assertion
Ref Expression
ovid ((x 𝑅 y 𝑆) → ((x𝐹y) = zφ))
Distinct variable groups:   x,y,z   z,𝑅   z,𝑆
Allowed substitution hints:   φ(x,y,z)   𝑅(x,y)   𝑆(x,y)   𝐹(x,y,z)

Proof of Theorem ovid
StepHypRef Expression
1 df-ov 5458 . . 3 (x𝐹y) = (𝐹‘⟨x, y⟩)
21eqeq1i 2044 . 2 ((x𝐹y) = z ↔ (𝐹‘⟨x, y⟩) = z)
3 ovid.1 . . . . . 6 ((x 𝑅 y 𝑆) → ∃!zφ)
43fnoprab 5546 . . . . 5 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)}
5 ovid.2 . . . . . 6 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}
65fneq1i 4936 . . . . 5 (𝐹 Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)} ↔ {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)})
74, 6mpbir 134 . . . 4 𝐹 Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)}
8 opabid 3985 . . . . 5 (⟨x, y {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)} ↔ (x 𝑅 y 𝑆))
98biimpri 124 . . . 4 ((x 𝑅 y 𝑆) → ⟨x, y {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)})
10 fnopfvb 5158 . . . 4 ((𝐹 Fn {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)} x, y {⟨x, y⟩ ∣ (x 𝑅 y 𝑆)}) → ((𝐹‘⟨x, y⟩) = z ↔ ⟨⟨x, y⟩, z 𝐹))
117, 9, 10sylancr 393 . . 3 ((x 𝑅 y 𝑆) → ((𝐹‘⟨x, y⟩) = z ↔ ⟨⟨x, y⟩, z 𝐹))
125eleq2i 2101 . . . . 5 (⟨⟨x, y⟩, z 𝐹 ↔ ⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)})
13 oprabid 5480 . . . . 5 (⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)} ↔ ((x 𝑅 y 𝑆) φ))
1412, 13bitri 173 . . . 4 (⟨⟨x, y⟩, z 𝐹 ↔ ((x 𝑅 y 𝑆) φ))
1514baib 827 . . 3 ((x 𝑅 y 𝑆) → (⟨⟨x, y⟩, z 𝐹φ))
1611, 15bitrd 177 . 2 ((x 𝑅 y 𝑆) → ((𝐹‘⟨x, y⟩) = zφ))
172, 16syl5bb 181 1 ((x 𝑅 y 𝑆) → ((x𝐹y) = zφ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = 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-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-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-v 2553  df-sbc 2759  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-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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