Theorem dffun2 6443

Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 11-Nov-2024.)

Assertion

Ref	Expression
dffun2	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun2

Step	Hyp	Ref	Expression
1		df-fun 6435	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
2		relco 6148	. . . . 5 ⊢ Rel (𝐴 ∘ ◡𝐴)
3		ssrel 5693	. . . . 5 ⊢ (Rel (𝐴 ∘ ◡𝐴) → ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑧(⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ◡𝐴) → ⟨𝑦, 𝑧⟩ ∈ I )))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑧(⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ◡𝐴) → ⟨𝑦, 𝑧⟩ ∈ I ))
5		vex 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
6		vex 3436	. . . . . . . 8 ⊢ 𝑧 ∈ V
7	5, 6	opelco 5780	. . . . . . 7 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ◡𝐴) ↔ ∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧))
8		vex 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9	5, 8	brcnv 5791	. . . . . . . . 9 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
10	9	anbi1i 624	. . . . . . . 8 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
11	10	exbii 1850	. . . . . . 7 ⊢ (∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
12	7, 11	bitri 274	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ◡𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
13		df-br 5075	. . . . . . 7 ⊢ (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
14	6	ideq 5761	. . . . . . 7 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
15	13, 14	bitr3i 276	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑦 = 𝑧)
16	12, 15	imbi12i 351	. . . . 5 ⊢ ((⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ◡𝐴) → ⟨𝑦, 𝑧⟩ ∈ I ) ↔ (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
17	16	2albii 1823	. . . 4 ⊢ (∀𝑦∀𝑧(⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ◡𝐴) → ⟨𝑦, 𝑧⟩ ∈ I ) ↔ ∀𝑦∀𝑧(∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
18		alrot3 2157	. . . . 5 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑧∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
19		19.23v 1945	. . . . . 6 ⊢ (∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
20	19	2albii 1823	. . . . 5 ⊢ (∀𝑦∀𝑧∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑧(∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
21	18, 20	bitr2i 275	. . . 4 ⊢ (∀𝑦∀𝑧(∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
22	4, 17, 21	3bitri 297	. . 3 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
23	22	anbi2i 623	. 2 ⊢ ((Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
24	1, 23	bitri 274	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074   I cid 5488  ^◡ccnv 5588   ∘ ccom 5593  Rel wrel 5594  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-fun 6435

This theorem is referenced by:  dffun3  6445  dffun4  6446  fundif  6483  fliftfun  7183  frrlem9  8110  fprlem1  8116  wfrlem5OLD  8144  wfrfunOLD  8150  frrlem15  9515  fpwwe2lem10  10396  fclim  15262  invfun  17476  lmfun  22532  ulmdm  25552  fundmpss  33740  fununiq  33743  fnsingle  34221  funimage  34230  funpartfun  34245