Theorem dffun2OLDOLD 6543

Description: Obsolete version of dffun2 6541 as of 11-Dec-2024. (Contributed by NM, 29-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dffun2OLDOLD	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun2OLDOLD

Step	Hyp	Ref	Expression
1		df-fun 6533	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
2		df-id 5548	. . . . . 6 ⊢ I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
3	2	sseq2i 3988	. . . . 5 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ (𝐴 ∘ ◡𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
4		df-co 5663	. . . . . 6 ⊢ (𝐴 ∘ ◡𝐴) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧)}
5	4	sseq1i 3987	. . . . 5 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
6		ssopab2bw 5522	. . . . 5 ⊢ ({⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ ∀𝑦∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
7	3, 5, 6	3bitri 297	. . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑦∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
8		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
9		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
10	8, 9	brcnv 5862	. . . . . . . . . . 11 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
11	10	anbi1i 624	. . . . . . . . . 10 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
12	11	exbii 1848	. . . . . . . . 9 ⊢ (∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧))
13	12	imbi1i 349	. . . . . . . 8 ⊢ ((∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
14		19.23v 1942	. . . . . . . 8 ⊢ (∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
15	13, 14	bitr4i 278	. . . . . . 7 ⊢ ((∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
16	15	albii 1819	. . . . . 6 ⊢ (∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
17		alcom 2159	. . . . . 6 ⊢ (∀𝑧∀𝑥((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
18	16, 17	bitri 275	. . . . 5 ⊢ (∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
19	18	albii 1819	. . . 4 ⊢ (∀𝑦∀𝑧(∃𝑥(𝑦◡𝐴𝑥 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
20		alcom 2159	. . . 4 ⊢ (∀𝑦∀𝑥∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
21	7, 19, 20	3bitri 297	. . 3 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ I ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))
22	21	anbi2i 623	. 2 ⊢ ((Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
23	1, 22	bitri 275	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779   ⊆ wss 3926   class class class wbr 5119  {copab 5181   I cid 5547  ^◡ccnv 5653   ∘ ccom 5658  Rel wrel 5659  Fun wfun 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-cnv 5662  df-co 5663  df-fun 6533

This theorem is referenced by: (None)