Theorem dffrege115 39665

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 2093	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
2		19.21v 1898	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
3		impexp 443	. . . . . . . . 9 ⊢ (((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)))
4		vex 3415	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
5		vex 3415	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
6	4, 5	brcnv 5600	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ 𝑐◡𝑅𝑏)
7		df-br 4928	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
8	5, 4	brcnv 5600	. . . . . . . . . . . 12 ⊢ (𝑐◡𝑅𝑏 ↔ 𝑏𝑅𝑐)
9	6, 7, 8	3bitr3ri 294	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
10		vex 3415	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
11	4, 10	brcnv 5600	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ 𝑎◡𝑅𝑏)
12		df-br 4928	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
13	10, 4	brcnv 5600	. . . . . . . . . . . 12 ⊢ (𝑎◡𝑅𝑏 ↔ 𝑏𝑅𝑎)
14	11, 12, 13	3bitr3ri 294	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
15	9, 14	anbi12ci 618	. . . . . . . . . 10 ⊢ ((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) ↔ (⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅))
16	15	imbi1i 342	. . . . . . . . 9 ⊢ (((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ ((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
17	3, 16	bitr3i 269	. . . . . . . 8 ⊢ ((𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
18	17	albii 1782	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
19	2, 18	bitr3i 269	. . . . . 6 ⊢ ((𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
20	19	albii 1782	. . . . 5 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑐∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
21		alcom 2093	. . . . 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 267	. . . 4 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
23		opeq2 4676	. . . . . 6 ⊢ (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
24	23	eleq1d 2847	. . . . 5 ⊢ (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅))
25	24	mo4 2578	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
26		df-mo 2544	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
27	22, 25, 26	3bitr2i 291	. . 3 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
28	27	albii 1782	. 2 ⊢ (∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
29		relcnv 5805	. . . 4 ⊢ Rel ◡◡𝑅
30	29	biantrur 523	. . 3 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
31		dffun5 6199	. . 3 ⊢ (Fun ◡◡𝑅 ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
32	30, 31	bitr4i 270	. 2 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ Fun ◡◡𝑅)
33	1, 28, 32	3bitri 289	1 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505  ∃wex 1742   ∈ wcel 2048  ∃*wmo 2542  ⟨cop 4445   class class class wbr 4927  ^◡ccnv 5403  Rel wrel 5409  Fun wfun 6180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pr 5184

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ral 3090  df-rab 3094  df-v 3414  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-sn 4440  df-pr 4442  df-op 4446  df-br 4928  df-opab 4990  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-fun 6188

This theorem is referenced by:  frege116  39666