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Theorem dffrege115 38589
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 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
Distinct variable group:   𝑎,𝑏,𝑐,𝑅

Proof of Theorem dffrege115
StepHypRef Expression
1 alcom 2077 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
2 19.21v 1908 . . . . . . 7 (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
3 impexp 461 . . . . . . . . 9 (((𝑏𝑅𝑐𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)))
4 vex 3234 . . . . . . . . . . . . 13 𝑏 ∈ V
5 vex 3234 . . . . . . . . . . . . 13 𝑐 ∈ V
64, 5brcnv 5337 . . . . . . . . . . . 12 (𝑏𝑅𝑐𝑐𝑅𝑏)
7 df-br 4686 . . . . . . . . . . . 12 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
85, 4brcnv 5337 . . . . . . . . . . . 12 (𝑐𝑅𝑏𝑏𝑅𝑐)
96, 7, 83bitr3ri 291 . . . . . . . . . . 11 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
10 vex 3234 . . . . . . . . . . . . 13 𝑎 ∈ V
114, 10brcnv 5337 . . . . . . . . . . . 12 (𝑏𝑅𝑎𝑎𝑅𝑏)
12 df-br 4686 . . . . . . . . . . . 12 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
1310, 4brcnv 5337 . . . . . . . . . . . 12 (𝑎𝑅𝑏𝑏𝑅𝑎)
1411, 12, 133bitr3ri 291 . . . . . . . . . . 11 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
159, 14anbi12ci 734 . . . . . . . . . 10 ((𝑏𝑅𝑐𝑏𝑅𝑎) ↔ (⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅))
1615imbi1i 338 . . . . . . . . 9 (((𝑏𝑅𝑐𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ ((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
173, 16bitr3i 266 . . . . . . . 8 ((𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
1817albii 1787 . . . . . . 7 (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
192, 18bitr3i 266 . . . . . 6 ((𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
2019albii 1787 . . . . 5 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑐𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
21 alcom 2077 . . . . 5 (∀𝑐𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐) ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
2220, 21bitri 264 . . . 4 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
23 opeq2 4434 . . . . . 6 (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
2423eleq1d 2715 . . . . 5 (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ 𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅))
2524mo4 2546 . . . 4 (∃*𝑎𝑏, 𝑎⟩ ∈ 𝑅 ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
26 mo2v 2505 . . . 4 (∃*𝑎𝑏, 𝑎⟩ ∈ 𝑅 ↔ ∃𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
2722, 25, 263bitr2i 288 . . 3 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∃𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
2827albii 1787 . 2 (∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
29 relcnv 5538 . . . 4 Rel 𝑅
3029biantrur 526 . . 3 (∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐) ↔ (Rel 𝑅 ∧ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐)))
31 dffun5 5939 . . 3 (Fun 𝑅 ↔ (Rel 𝑅 ∧ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐)))
3230, 31bitr4i 267 . 2 (∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐) ↔ Fun 𝑅)
331, 28, 323bitri 286 1 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744  wcel 2030  ∃*wmo 2499  cop 4216   class class class wbr 4685  ccnv 5142  Rel wrel 5148  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-fun 5928
This theorem is referenced by:  frege116  38590
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