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Theorem fun2cnv 4963
 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 4959 . 2 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑦𝐴𝑥)
2 vex 2560 . . . . 5 𝑦 ∈ V
3 vex 2560 . . . . 5 𝑥 ∈ V
42, 3brcnv 4518 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
54mobii 1937 . . 3 (∃*𝑦 𝑦𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦)
65albii 1359 . 2 (∀𝑥∃*𝑦 𝑦𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
71, 6bitri 173 1 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1241  ∃*wmo 1901   class class class wbr 3764  ◡ccnv 4344  Fun wfun 4896 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-fun 4904 This theorem is referenced by:  svrelfun  4964  fun11  4966
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