Theorem svrelfun 5323

Description: A single-valued relation is a function. (See fun2cnv 5322 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.

Step	Hyp	Ref	Expression
1		dffun6 5272	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2		fun2cnv 5322	. . 3 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
3	2	anbi2i 457	. 2 ⊢ ((Rel 𝐴 ∧ Fun ◡◡𝐴) ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4	1, 3	bitr4i 187	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun ◡◡𝐴))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1362  ∃*wmo 2046   class class class wbr 4033  ^◡ccnv 4662  Rel wrel 4668  Fun wfun 5252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-fun 5260

This theorem is referenced by: (None)