Theorem dffun3 6335

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

Assertion

Ref	Expression
dffun3	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem dffun3

Step	Hyp	Ref	Expression
1		dffun2 6334	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
2		breq2 5034	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐴𝑦 ↔ 𝑥𝐴𝑧))
3	2	mo4 2625	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
4		df-mo 2598	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
5	3, 4	bitr3i 280	. . . 4 ⊢ (∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
6	5	albii 1821	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
7	6	anbi2i 625	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
8	1, 7	bitri 278	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  ∃*wmo 2596   class class class wbr 5030  Rel wrel 5524  Fun wfun 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-cnv 5527  df-co 5528  df-fun 6326

This theorem is referenced by:  dffun5  6337  dffun6f  6338  sbcfung  6348  dffv2  6733