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Theorem funcnv3 4904
 Description: A condition showing a class is single-rooted. (See funcnv 4903). (Contributed by NM, 26-May-2006.)
Assertion
Ref Expression
funcnv3 (Fun Ay ran A∃!x dom A xAy)
Distinct variable group:   x,y,A

Proof of Theorem funcnv3
StepHypRef Expression
1 dfrn2 4466 . . . . . 6 ran A = {yx xAy}
21abeq2i 2145 . . . . 5 (y ran Ax xAy)
32biimpi 113 . . . 4 (y ran Ax xAy)
43biantrurd 289 . . 3 (y ran A → (∃*x xAy ↔ (x xAy ∃*x xAy)))
54ralbiia 2332 . 2 (y ran A∃*x xAyy ran A(x xAy ∃*x xAy))
6 funcnv 4903 . 2 (Fun Ay ran A∃*x xAy)
7 df-reu 2307 . . . 4 (∃!x dom A xAy∃!x(x dom A xAy))
8 vex 2554 . . . . . . 7 x V
9 vex 2554 . . . . . . 7 y V
108, 9breldm 4482 . . . . . 6 (xAyx dom A)
1110pm4.71ri 372 . . . . 5 (xAy ↔ (x dom A xAy))
1211eubii 1906 . . . 4 (∃!x xAy∃!x(x dom A xAy))
13 eu5 1944 . . . 4 (∃!x xAy ↔ (x xAy ∃*x xAy))
147, 12, 133bitr2i 197 . . 3 (∃!x dom A xAy ↔ (x xAy ∃*x xAy))
1514ralbii 2324 . 2 (y ran A∃!x dom A xAyy ran A(x xAy ∃*x xAy))
165, 6, 153bitr4i 201 1 (Fun Ay ran A∃!x dom A xAy)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∃*wmo 1898  ∀wral 2300  ∃!wreu 2302   class class class wbr 3755  ◡ccnv 4287  dom cdm 4288  ran crn 4289  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-reu 2307  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-dm 4298  df-rn 4299  df-fun 4847 This theorem is referenced by: (None)
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