Theorem dffrege115 41586

Description: If from the circumstance that 𝑐 is a result of an application of the procedure 𝑅 to 𝑏, whatever 𝑏 may be, it can be inferred that every result of an application of the procedure 𝑅 to 𝑏 is the same as 𝑐, then we say : "The procedure 𝑅 is single-valued". Definition 115 of [Frege1879] p. 77. (Contributed by RP, 7-Jul-2020.)

Assertion

Ref	Expression
dffrege115	⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)

Distinct variable group:   𝑎,𝑏,𝑐,𝑅

Proof of Theorem dffrege115

Step	Hyp	Ref	Expression
1		alcom 2156	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
2		19.21v 1942	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
3		impexp 451	. . . . . . . . 9 ⊢ (((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)))
4		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
5		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
6	4, 5	brcnv 5791	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ 𝑐◡𝑅𝑏)
7		df-br 5075	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
8	5, 4	brcnv 5791	. . . . . . . . . . . 12 ⊢ (𝑐◡𝑅𝑏 ↔ 𝑏𝑅𝑐)
9	6, 7, 8	3bitr3ri 302	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
10		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
11	4, 10	brcnv 5791	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ 𝑎◡𝑅𝑏)
12		df-br 5075	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
13	10, 4	brcnv 5791	. . . . . . . . . . . 12 ⊢ (𝑎◡𝑅𝑏 ↔ 𝑏𝑅𝑎)
14	11, 12, 13	3bitr3ri 302	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
15	9, 14	anbi12ci 628	. . . . . . . . . 10 ⊢ ((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) ↔ (⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅))
16	15	imbi1i 350	. . . . . . . . 9 ⊢ (((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ ((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
17	3, 16	bitr3i 276	. . . . . . . 8 ⊢ ((𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
18	17	albii 1822	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
19	2, 18	bitr3i 276	. . . . . 6 ⊢ ((𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
20	19	albii 1822	. . . . 5 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑐∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
21		alcom 2156	. . . . 5 ⊢ (∀𝑐∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐) ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
22	20, 21	bitri 274	. . . 4 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
23		opeq2 4805	. . . . . 6 ⊢ (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
24	23	eleq1d 2823	. . . . 5 ⊢ (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅))
25	24	mo4 2566	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
26		df-mo 2540	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
27	22, 25, 26	3bitr2i 299	. . 3 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
28	27	albii 1822	. 2 ⊢ (∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
29		relcnv 6012	. . . 4 ⊢ Rel ◡◡𝑅
30	29	biantrur 531	. . 3 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
31		dffun5 6447	. . 3 ⊢ (Fun ◡◡𝑅 ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
32	30, 31	bitr4i 277	. 2 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ Fun ◡◡𝑅)
33	1, 28, 32	3bitri 297	1 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782   ∈ wcel 2106  ∃*wmo 2538  ⟨cop 4567   class class class wbr 5074  ^◡ccnv 5588  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-mo 2540  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:  frege116  41587