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Theorem funcnv2 6636
Description: A simpler equivalence for single-rooted (see funcnv 6637). (Contributed by NM, 9-Aug-2004.)
Assertion
Ref Expression
funcnv2 (Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem funcnv2
StepHypRef Expression
1 relcnv 6125 . . 3 Rel 𝐴
2 dffun6 6576 . . 3 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑦∃*𝑥 𝑦𝐴𝑥))
31, 2mpbiran 709 . 2 (Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑦𝐴𝑥)
4 vex 3482 . . . . 5 𝑦 ∈ V
5 vex 3482 . . . . 5 𝑥 ∈ V
64, 5brcnv 5896 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
76mobii 2546 . . 3 (∃*𝑥 𝑦𝐴𝑥 ↔ ∃*𝑥 𝑥𝐴𝑦)
87albii 1816 . 2 (∀𝑦∃*𝑥 𝑦𝐴𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
93, 8bitri 275 1 (Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1535  ∃*wmo 2536   class class class wbr 5148  ccnv 5688  Rel wrel 5694  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by:  funcnv  6637  fun2cnv  6639  fun11  6642  dff12  6804  1stconst  8124  2ndconst  8125
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