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Theorem funcnv 4991
Description: The converse of a class is a function iff the class is single-rooted, which means that for any 𝑦 in the range of 𝐴 there is at most one 𝑥 such that 𝑥𝐴𝑦. Definition of single-rooted in [Enderton] p. 43. See funcnv2 4990 for a simpler version. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
funcnv (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem funcnv
StepHypRef Expression
1 vex 2605 . . . . . . 7 𝑥 ∈ V
2 vex 2605 . . . . . . 7 𝑦 ∈ V
31, 2brelrn 4595 . . . . . 6 (𝑥𝐴𝑦𝑦 ∈ ran 𝐴)
43pm4.71ri 384 . . . . 5 (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
54mobii 1979 . . . 4 (∃*𝑥 𝑥𝐴𝑦 ↔ ∃*𝑥(𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
6 moanimv 2017 . . . 4 (∃*𝑥(𝑦 ∈ ran 𝐴𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
75, 6bitri 182 . . 3 (∃*𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
87albii 1400 . 2 (∀𝑦∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
9 funcnv2 4990 . 2 (Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10 df-ral 2354 . 2 (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
118, 9, 103bitr4i 210 1 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wcel 1434  ∃*wmo 1943  wral 2349   class class class wbr 3793  ccnv 4370  ran crn 4372  Fun wfun 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934
This theorem is referenced by:  funcnv3  4992  fncnv  4996
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