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Theorem ovig 5564
Description: The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovig.1 ((x = A y = B z = 𝐶) → (φψ))
ovig.2 ((x 𝑅 y 𝑆) → ∃*zφ)
ovig.3 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}
Assertion
Ref Expression
ovig ((A 𝑅 B 𝑆 𝐶 𝐷) → (ψ → (A𝐹B) = 𝐶))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   x,𝑅,y,z   x,𝑆,y,z   ψ,x,y,z
Allowed substitution hints:   φ(x,y,z)   𝐷(x,y,z)   𝐹(x,y,z)

Proof of Theorem ovig
StepHypRef Expression
1 3simpa 900 . 2 ((A 𝑅 B 𝑆 𝐶 𝐷) → (A 𝑅 B 𝑆))
2 eleq1 2097 . . . . . 6 (x = A → (x 𝑅A 𝑅))
3 eleq1 2097 . . . . . 6 (y = B → (y 𝑆B 𝑆))
42, 3bi2anan9 538 . . . . 5 ((x = A y = B) → ((x 𝑅 y 𝑆) ↔ (A 𝑅 B 𝑆)))
543adant3 923 . . . 4 ((x = A y = B z = 𝐶) → ((x 𝑅 y 𝑆) ↔ (A 𝑅 B 𝑆)))
6 ovig.1 . . . 4 ((x = A y = B z = 𝐶) → (φψ))
75, 6anbi12d 442 . . 3 ((x = A y = B z = 𝐶) → (((x 𝑅 y 𝑆) φ) ↔ ((A 𝑅 B 𝑆) ψ)))
8 ovig.2 . . . 4 ((x 𝑅 y 𝑆) → ∃*zφ)
9 moanimv 1972 . . . 4 (∃*z((x 𝑅 y 𝑆) φ) ↔ ((x 𝑅 y 𝑆) → ∃*zφ))
108, 9mpbir 134 . . 3 ∃*z((x 𝑅 y 𝑆) φ)
11 ovig.3 . . 3 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑅 y 𝑆) φ)}
127, 10, 11ovigg 5563 . 2 ((A 𝑅 B 𝑆 𝐶 𝐷) → (((A 𝑅 B 𝑆) ψ) → (A𝐹B) = 𝐶))
131, 12mpand 405 1 ((A 𝑅 B 𝑆 𝐶 𝐷) → (ψ → (A𝐹B) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  ∃*wmo 1898  (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-bndl 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-fv 4853  df-ov 5458  df-oprab 5459
This theorem is referenced by:  th3q  6147  addnnnq0  6432  mulnnnq0  6433  addsrpr  6673  mulsrpr  6674
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