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Theorem fun2cnv 6414
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 6411 . 2 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑦𝐴𝑥)
2 vex 3484 . . . . 5 𝑦 ∈ V
3 vex 3484 . . . . 5 𝑥 ∈ V
42, 3brcnv 5741 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
54mobii 2632 . . 3 (∃*𝑦 𝑦𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦)
65albii 1821 . 2 (∀𝑥∃*𝑦 𝑦𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
71, 6bitri 278 1 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  ∃*wmo 2622   class class class wbr 5053  ccnv 5542  Fun wfun 6338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3483  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-fun 6346
This theorem is referenced by:  svrelfun  6415  fun11  6417
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