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Theorem fun2cnv 4991
Description: The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that 𝐴 is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
fun2cnv (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem fun2cnv
StepHypRef Expression
1 funcnv2 4987 . 2 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑦𝐴𝑥)
2 vex 2577 . . . . 5 𝑦 ∈ V
3 vex 2577 . . . . 5 𝑥 ∈ V
42, 3brcnv 4546 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
54mobii 1953 . . 3 (∃*𝑦 𝑦𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦)
65albii 1375 . 2 (∀𝑥∃*𝑦 𝑦𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
71, 6bitri 177 1 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257  ∃*wmo 1917   class class class wbr 3792  ccnv 4372  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-fun 4932
This theorem is referenced by:  svrelfun  4992  fun11  4994
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