Theorem dffrege115 40317

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 2159	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
2		19.21v 1936	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
3		impexp 453	. . . . . . . . 9 ⊢ (((𝑏𝑅𝑐 ∧ 𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)))
4		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
5		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
6	4, 5	brcnv 5747	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ 𝑐◡𝑅𝑏)
7		df-br 5059	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
8	5, 4	brcnv 5747	. . . . . . . . . . . 12 ⊢ (𝑐◡𝑅𝑏 ↔ 𝑏𝑅𝑐)
9	6, 7, 8	3bitr3ri 304	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅)
10		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
11	4, 10	brcnv 5747	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ 𝑎◡𝑅𝑏)
12		df-br 5059	. . . . . . . . . . . 12 ⊢ (𝑏◡◡𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
13	10, 4	brcnv 5747	. . . . . . . . . . . 12 ⊢ (𝑎◡𝑅𝑏 ↔ 𝑏𝑅𝑎)
14	11, 12, 13	3bitr3ri 304	. . . . . . . . . . 11 ⊢ (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅)
15	9, 14	anbi12ci 629	. . . . . . . . . 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 279	. . . . . . . 8 ⊢ ((𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
18	17	albii 1816	. . . . . . 7 ⊢ (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
19	2, 18	bitr3i 279	. . . . . 6 ⊢ ((𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
20	19	albii 1816	. . . . 5 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑐∀𝑎((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
21		alcom 2159	. . . . 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 277	. . . 4 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
23		opeq2 4797	. . . . . 6 ⊢ (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
24	23	eleq1d 2897	. . . . 5 ⊢ (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅))
25	24	mo4 2646	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∀𝑎∀𝑐((⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ ◡◡𝑅) → 𝑎 = 𝑐))
26		df-mo 2618	. . . 4 ⊢ (∃*𝑎⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
27	22, 25, 26	3bitr2i 301	. . 3 ⊢ (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
28	27	albii 1816	. 2 ⊢ (∀𝑏∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐))
29		relcnv 5961	. . . 4 ⊢ Rel ◡◡𝑅
30	29	biantrur 533	. . 3 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
31		dffun5 6362	. . 3 ⊢ (Fun ◡◡𝑅 ↔ (Rel ◡◡𝑅 ∧ ∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐)))
32	30, 31	bitr4i 280	. 2 ⊢ (∀𝑏∃𝑐∀𝑎(⟨𝑏, 𝑎⟩ ∈ ◡◡𝑅 → 𝑎 = 𝑐) ↔ Fun ◡◡𝑅)
33	1, 28, 32	3bitri 299	1 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  ∃wex 1776   ∈ wcel 2110  ∃*wmo 2616  ⟨cop 4566   class class class wbr 5058  ^◡ccnv 5548  Rel wrel 5554  Fun wfun 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-fun 6351

This theorem is referenced by:  frege116  40318