Theorem dffun5 6579

Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Assertion

Ref	Expression
dffun5	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun5

Step	Hyp	Ref	Expression
1		dffun3 6576	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
2		df-br 5148	. . . . . . 7 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	2	imbi1i 349	. . . . . 6 ⊢ ((𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))
4	3	albii 1815	. . . . 5 ⊢ (∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))
5	4	exbii 1844	. . . 4 ⊢ (∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))
6	5	albii 1815	. . 3 ⊢ (∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))
7	6	anbi2i 623	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))
8	1, 7	bitri 275	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1534  ∃wex 1775   ∈ wcel 2105  ⟨cop 4636   class class class wbr 5147  Rel wrel 5693  Fun wfun 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-fun 6564

This theorem is referenced by:  funimaexgOLD  6654  fvn0ssdmfun  7093  uzrdgfni  13995  noseqrdgfn  28326  dffrege115  43967