Theorem dffun3 5815

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 5814	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
2		breq2 4587	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐴𝑦 ↔ 𝑥𝐴𝑧))
3	2	mo4 2505	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
4		mo2v 2465	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
5	3, 4	bitr3i 265	. . . 4 ⊢ (∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
6	5	albii 1737	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
7	6	anbi2i 726	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
8	1, 7	bitri 263	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  ∃*wmo 2459   class class class wbr 4583  Rel wrel 5043  Fun wfun 5798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-cnv 5046  df-co 5047  df-fun 5806

This theorem is referenced by:  dffun5  5817  dffun6f  5818  sbcfung  5827  dffv2  6181