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Theorem fun2cnv 5252
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 5248 . 2 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑦𝐴𝑥)
2 vex 2729 . . . . 5 𝑦 ∈ V
3 vex 2729 . . . . 5 𝑥 ∈ V
42, 3brcnv 4787 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
54mobii 2051 . . 3 (∃*𝑦 𝑦𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦)
65albii 1458 . 2 (∀𝑥∃*𝑦 𝑦𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
71, 6bitri 183 1 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1341  ∃*wmo 2015   class class class wbr 3982  ccnv 4603  Fun wfun 5182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-fun 5190
This theorem is referenced by:  svrelfun  5253  fun11  5255
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