Theorem svrelfun 6426

Description: A single-valued relation is a function. (See fun2cnv 6425 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 6370	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
2		fun2cnv 6425	. . 3 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
3	2	anbi2i 624	. 2 ⊢ ((Rel 𝐴 ∧ Fun ◡◡𝐴) ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4	1, 3	bitr4i 280	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun ◡◡𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃*wmo 2620   class class class wbr 5066  ^◡ccnv 5554  Rel wrel 5560  Fun wfun 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-fun 6357

This theorem is referenced by: (None)