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Theorem funcnveq 4905
Description: Another way of expressing that a class is single-rooted. Counterpart to dffun2 4855. (Contributed by Jim Kingdon, 24-Dec-2018.)
Assertion
Ref Expression
funcnveq (Fun Axyz((xAy zAy) → x = z))
Distinct variable group:   x,y,z,A

Proof of Theorem funcnveq
StepHypRef Expression
1 relcnv 4646 . . 3 Rel A
2 dffun2 4855 . . 3 (Fun A ↔ (Rel A yxz((yAx yAz) → x = z)))
31, 2mpbiran 846 . 2 (Fun Ayxz((yAx yAz) → x = z))
4 alcom 1364 . 2 (yxz((yAx yAz) → x = z) ↔ xyz((yAx yAz) → x = z))
5 vex 2554 . . . . . . 7 y V
6 vex 2554 . . . . . . 7 x V
75, 6brcnv 4461 . . . . . 6 (yAxxAy)
8 vex 2554 . . . . . . 7 z V
95, 8brcnv 4461 . . . . . 6 (yAzzAy)
107, 9anbi12i 433 . . . . 5 ((yAx yAz) ↔ (xAy zAy))
1110imbi1i 227 . . . 4 (((yAx yAz) → x = z) ↔ ((xAy zAy) → x = z))
12112albii 1357 . . 3 (yz((yAx yAz) → x = z) ↔ yz((xAy zAy) → x = z))
1312albii 1356 . 2 (xyz((yAx yAz) → x = z) ↔ xyz((xAy zAy) → x = z))
143, 4, 133bitri 195 1 (Fun Axyz((xAy zAy) → x = z))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   class class class wbr 3755  ccnv 4287  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-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-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-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847
This theorem is referenced by:  imain  4924
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