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Theorem dffrege115 38793
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 2187 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
2 19.21v 2018 . . . . . . 7 (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
3 impexp 461 . . . . . . . . 9 (((𝑏𝑅𝑐𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)))
4 vex 3344 . . . . . . . . . . . . 13 𝑏 ∈ V
5 vex 3344 . . . . . . . . . . . . 13 𝑐 ∈ V
64, 5brcnv 5461 . . . . . . . . . . . 12 (𝑏𝑅𝑐𝑐𝑅𝑏)
7 df-br 4806 . . . . . . . . . . . 12 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
85, 4brcnv 5461 . . . . . . . . . . . 12 (𝑐𝑅𝑏𝑏𝑅𝑐)
96, 7, 83bitr3ri 291 . . . . . . . . . . 11 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
10 vex 3344 . . . . . . . . . . . . 13 𝑎 ∈ V
114, 10brcnv 5461 . . . . . . . . . . . 12 (𝑏𝑅𝑎𝑎𝑅𝑏)
12 df-br 4806 . . . . . . . . . . . 12 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
1310, 4brcnv 5461 . . . . . . . . . . . 12 (𝑎𝑅𝑏𝑏𝑅𝑎)
1411, 12, 133bitr3ri 291 . . . . . . . . . . 11 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
159, 14anbi12ci 736 . . . . . . . . . 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 1896 . . . . . . 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 1896 . . . . 5 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑐𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
21 alcom 2187 . . . . 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 4555 . . . . . 6 (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
2423eleq1d 2825 . . . . 5 (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ 𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅))
2524mo4 2656 . . . 4 (∃*𝑎𝑏, 𝑎⟩ ∈ 𝑅 ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
26 mo2v 2615 . . . 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 1896 . 2 (∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
29 relcnv 5662 . . . 4 Rel 𝑅
3029biantrur 528 . . 3 (∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐) ↔ (Rel 𝑅 ∧ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐)))
31 dffun5 6063 . . 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 1630  wex 1853  wcel 2140  ∃*wmo 2609  cop 4328   class class class wbr 4805  ccnv 5266  Rel wrel 5272  Fun wfun 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-fun 6052
This theorem is referenced by:  frege116  38794
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