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Theorem svrelfun 6397
 Description: A single-valued relation is a function. (See fun2cnv 6396 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
svrelfun (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun 𝐴))

Proof of Theorem svrelfun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun6 6340 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2 fun2cnv 6396 . . 3 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
32anbi2i 625 . 2 ((Rel 𝐴 ∧ Fun 𝐴) ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
41, 3bitr4i 281 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃*wmo 2596   class class class wbr 5031  ◡ccnv 5519  Rel wrel 5525  Fun wfun 6319 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-fun 6327 This theorem is referenced by: (None)
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