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Theorem svrelfun 5949
Description: A single-valued relation is a function. (See fun2cnv 5948 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 5891 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2 fun2cnv 5948 . . 3 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
32anbi2i 729 . 2 ((Rel 𝐴 ∧ Fun 𝐴) ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
41, 3bitr4i 267 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wal 1479  ∃*wmo 2469   class class class wbr 4644  ccnv 5103  Rel wrel 5109  Fun wfun 5870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-fun 5878
This theorem is referenced by: (None)
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