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Theorem svrelfun 5261
Description: A single-valued relation is a function. (See fun2cnv 5260 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 5210 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2 fun2cnv 5260 . . 3 (Fun 𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
32anbi2i 454 . 2 ((Rel 𝐴 ∧ Fun 𝐴) ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
41, 3bitr4i 186 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun 𝐴))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1346  ∃*wmo 2020   class class class wbr 3987  ccnv 4608  Rel wrel 4614  Fun wfun 5190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-fun 5198
This theorem is referenced by: (None)
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