Theorem dffrege115 45168

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 2199	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
2		19.21v 1974	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
3		impexp 456	. . . . . . . . 9 ⊢ (((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)))
4		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
5		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
6	4, 5	brcnv 5866	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ 𝑐◡𝑅𝑏)
7		df-br 5107	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
8	5, 4	brcnv 5866	. . . . . . . . . . . 12 ⊢ (𝑐◡𝑅𝑏 ↔ 𝑏𝑅𝑐)
9	6, 7, 8	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
10		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
11	4, 10	brcnv 5866	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ 𝑎◡𝑅𝑏)
12		df-br 5107	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
13	10, 4	brcnv 5866	. . . . . . . . . . . 12 ⊢ (𝑎◡𝑅𝑏 ↔ 𝑏𝑅𝑎)
14	11, 12, 13	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
15	9, 14	anbi12ci 641	. . . . . . . . . 10 ⊢ ((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) ↔ (⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅))
16	15	imbi1i 352	. . . . . . . . 9 ⊢ (((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ ((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
17	3, 16	bitr3i 280	. . . . . . . 8 ⊢ ((𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
18	17	albii 1852	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
19	2, 18	bitr3i 280	. . . . . 6 ⊢ ((𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
20	19	albii 1852	. . . . 5 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑐∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
21		alcom 2199	. . . . 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 278	. . . 4 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
23		opeq2 4844	. . . . . 6 ⊢ (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
24	23	eleq1d 2849	. . . . 5 ⊢ (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅))
25	24	mo4 2594	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
26		dfmo 2568	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
27	22, 25, 26	3bitr2i 302	. . 3 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
28	27	albii 1852	. 2 ⊢ (∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
29		relcnv 6111	. . . 4 ⊢ Rel ◡◡𝑅
30	29	biantrur 540	. . 3 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
31		dffun5 6569	. . 3 ⊢ (Fun ◡◡𝑅 ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
32	30, 31	bitr4i 281	. 2 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ Fun ◡◡𝑅)
33	1, 28, 32	3bitri 300	1 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2147  ∃*wmo 2565  ⟨cop 4838   class class class wbr 5106  ^◡ccnv 5654  Rel wrel 5660  Fun wfun 6549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-fun 6557

This theorem is used by:  frege116  45169