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Theorem funcnv3 4883
Description: A condition showing a class is single-rooted. (See funcnv 4882). (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 4446 . . . . . 6 ran A = {yx xAy}
21abeq2i 2126 . . . . 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 2312 . 2 (y ran A∃*x xAyy ran A(x xAy ∃*x xAy))
6 funcnv 4882 . 2 (Fun Ay ran A∃*x xAy)
7 df-reu 2287 . . . 4 (∃!x dom A xAy∃!x(x dom A xAy))
8 vex 2534 . . . . . . 7 x V
9 vex 2534 . . . . . . 7 y V
108, 9breldm 4462 . . . . . 6 (xAyx dom A)
1110pm4.71ri 372 . . . . 5 (xAy ↔ (x dom A xAy))
1211eubii 1887 . . . 4 (∃!x xAy∃!x(x dom A xAy))
13 eu5 1925 . . . 4 (∃!x xAy ↔ (x xAy ∃*x xAy))
147, 12, 133bitr2i 197 . . 3 (∃!x dom A xAy ↔ (x xAy ∃*x xAy))
1514ralbii 2304 . 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 1358   wcel 1370  ∃!weu 1878  ∃*wmo 1879  wral 2280  ∃!wreu 2282   class class class wbr 3734  ccnv 4267  dom cdm 4268  ran crn 4269  Fun wfun 4819
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-fun 4827
This theorem is referenced by: (None)
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