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Theorem th3qcor 6146
 Description: Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
th3q.1 V
th3q.2 Er (𝑆 × 𝑆)
th3q.4 ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) ((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆))) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)))
th3q.5 𝐺 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))}
Assertion
Ref Expression
th3qcor Fun 𝐺
Distinct variable groups:   x,y,z,w,v,u,𝑡,𝑠,f,g,,   x,𝑆,y,z,w,v,u,𝑡,𝑠,f,g,   x, + ,y,z,w,v,u,𝑡,𝑠,f,g,
Allowed substitution hints:   𝐺(x,y,z,w,v,u,𝑡,f,g,,𝑠)

Proof of Theorem th3qcor
StepHypRef Expression
1 th3q.1 . . . . 5 V
2 th3q.2 . . . . 5 Er (𝑆 × 𝑆)
3 th3q.4 . . . . 5 ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) ((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆))) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)))
41, 2, 3th3qlem2 6145 . . . 4 ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) → ∃*zwvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))
5 moanimv 1972 . . . 4 (∃*z((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] )) ↔ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) → ∃*zwvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] )))
64, 5mpbir 134 . . 3 ∃*z((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))
76funoprab 5543 . 2 Fun {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))}
8 th3q.5 . . 3 𝐺 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))}
98funeqi 4865 . 2 (Fun 𝐺 ↔ Fun {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))})
107, 9mpbir 134 1 Fun 𝐺
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  Vcvv 2551  ⟨cop 3370   class class class wbr 3755   × cxp 4286  Fun wfun 4839  (class class class)co 5455  {coprab 5456   Er wer 6039  [cec 6040   / cqs 6041 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-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-er 6042  df-ec 6044  df-qs 6048 This theorem is referenced by: (None)
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