Theorem dffun2 5814

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

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem dffun2

Step	Hyp	Ref	Expression
1		df-fun 5806	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
2		df-id 4953	. . . . . 6 ⊢ I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
3	2	sseq2i 3593	. . . . 5 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ (𝐴 ∘ ◡𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
4		df-co 5047	. . . . . 6 ⊢ (𝐴 ∘ ◡𝐴) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧)}
5	4	sseq1i 3592	. . . . 5 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
6		ssopab2b 4927	. . . . 5 ⊢ ({⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ ∀𝑦∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
7	3, 5, 6	3bitri 285	. . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
8		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
9		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
10	8, 9	brcnv 5227	. . . . . . . . . . 11 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
11	10	anbi1i 727	. . . . . . . . . 10 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
12	11	exbii 1764	. . . . . . . . 9 ⊢ (∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
13	12	imbi1i 338	. . . . . . . 8 ⊢ ((∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
14		19.23v 1889	. . . . . . . 8 ⊢ (∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
15	13, 14	bitr4i 266	. . . . . . 7 ⊢ ((∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
16	15	albii 1737	. . . . . 6 ⊢ (∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
17		alcom 2024	. . . . . 6 ⊢ (∀𝑧∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
18	16, 17	bitri 263	. . . . 5 ⊢ (∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
19	18	albii 1737	. . . 4 ⊢ (∀𝑦∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
20		alcom 2024	. . . 4 ⊢ (∀𝑦∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
21	7, 19, 20	3bitri 285	. . 3 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
22	21	anbi2i 726	. 2 ⊢ ((Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
23	1, 22	bitri 263	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695   ⊆ wss 3540   class class class wbr 4583  {copab 4642   I cid 4948  ^◡ccnv 5037   ∘ ccom 5042  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:  dffun3  5815  dffun4  5816  fundif  5849  fliftfun  6462  wfrlem5  7306  wfrfun  7312  fpwwe2lem11  9341  fclim  14132  invfun  16247  lmfun  20995  ulmdm  23951  fundmpss  30910  fununiq  30913  frrlem5  31028  frrlem5c  31030  fnsingle  31196  funimage  31205  funpartfun  31220