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Theorem funcnv2 6634
Description: A simpler equivalence for single-rooted (see funcnv 6635). (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 6122 . . 3 Rel 𝐴
2 dffun6 6574 . . 3 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑦∃*𝑥 𝑦𝐴𝑥))
31, 2mpbiran 709 . 2 (Fun 𝐴 ↔ ∀𝑦∃*𝑥 𝑦𝐴𝑥)
4 vex 3484 . . . . 5 𝑦 ∈ V
5 vex 3484 . . . . 5 𝑥 ∈ V
64, 5brcnv 5893 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
76mobii 2548 . . 3 (∃*𝑥 𝑦𝐴𝑥 ↔ ∃*𝑥 𝑥𝐴𝑦)
87albii 1819 . 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 1538  ∃*wmo 2538   class class class wbr 5143  ccnv 5684  Rel wrel 5690  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563
This theorem is referenced by:  funcnv  6635  fun2cnv  6637  fun11  6640  dff12  6803  1stconst  8125  2ndconst  8126
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