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Theorem dffun5r 4857
Description: A way of proving a relation is a function, analogous to mo2r 1949. (Contributed by Jim Kingdon, 27-May-2020.)
Assertion
Ref Expression
dffun5r ((Rel A xzy(⟨x, y Ay = z)) → Fun A)
Distinct variable group:   x,y,z,A

Proof of Theorem dffun5r
StepHypRef Expression
1 nfv 1418 . . . . . 6 zx, y A
21mo2r 1949 . . . . 5 (zy(⟨x, y Ay = z) → ∃*yx, y A)
3 opeq2 3541 . . . . . . 7 (y = z → ⟨x, y⟩ = ⟨x, z⟩)
43eleq1d 2103 . . . . . 6 (y = z → (⟨x, y A ↔ ⟨x, z A))
54mo4 1958 . . . . 5 (∃*yx, y Ayz((⟨x, y A x, z A) → y = z))
62, 5sylib 127 . . . 4 (zy(⟨x, y Ay = z) → yz((⟨x, y A x, z A) → y = z))
76alimi 1341 . . 3 (xzy(⟨x, y Ay = z) → xyz((⟨x, y A x, z A) → y = z))
87anim2i 324 . 2 ((Rel A xzy(⟨x, y Ay = z)) → (Rel A xyz((⟨x, y A x, z A) → y = z)))
9 dffun4 4856 . 2 (Fun A ↔ (Rel A xyz((⟨x, y A x, z A) → y = z)))
108, 9sylibr 137 1 ((Rel A xzy(⟨x, y Ay = z)) → Fun A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240  wex 1378   wcel 1390  ∃*wmo 1898  cop 3370  Rel wrel 4293  Fun wfun 4839
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-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-cnv 4296  df-co 4297  df-fun 4847
This theorem is referenced by: (None)
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