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Theorem oprabex3 5698
Description: Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
Hypotheses
Ref Expression
oprabex3.1 𝐻 V
oprabex3.2 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}
Assertion
Ref Expression
oprabex3 𝐹 V
Distinct variable groups:   x,y,z,w,v,u,f,𝐻   x,𝑅,y,z
Allowed substitution hints:   𝑅(w,v,u,f)   𝐹(x,y,z,w,v,u,f)

Proof of Theorem oprabex3
StepHypRef Expression
1 oprabex3.1 . . 3 𝐻 V
21, 1xpex 4396 . 2 (𝐻 × 𝐻) V
3 moeq 2710 . . . . . 6 ∃*z z = 𝑅
43mosubop 4349 . . . . 5 ∃*zuf(y = ⟨u, f z = 𝑅)
54mosubop 4349 . . . 4 ∃*zwv(x = ⟨w, v uf(y = ⟨u, f z = 𝑅))
6 anass 381 . . . . . . . 8 (((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ (x = ⟨w, v (y = ⟨u, f z = 𝑅)))
762exbii 1494 . . . . . . 7 (uf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ uf(x = ⟨w, v (y = ⟨u, f z = 𝑅)))
8 19.42vv 1785 . . . . . . 7 (uf(x = ⟨w, v (y = ⟨u, f z = 𝑅)) ↔ (x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
97, 8bitri 173 . . . . . 6 (uf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ (x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
1092exbii 1494 . . . . 5 (wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ wv(x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
1110mobii 1934 . . . 4 (∃*zwvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅) ↔ ∃*zwv(x = ⟨w, v uf(y = ⟨u, f z = 𝑅)))
125, 11mpbir 134 . . 3 ∃*zwvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅)
1312a1i 9 . 2 ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) → ∃*zwvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))
14 oprabex3.2 . 2 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝐻 × 𝐻) y (𝐻 × 𝐻)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝑅))}
152, 2, 13, 14oprabex 5697 1 𝐹 V
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  ∃*wmo 1898  Vcvv 2551  cop 3370   × cxp 4286  {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-13 1401  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  ax-un 4136
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-oprab 5459
This theorem is referenced by: (None)
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