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Theorem dffrege115 40610
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 2163 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
2 19.21v 1940 . . . . . . 7 (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
3 impexp 454 . . . . . . . . 9 (((𝑏𝑅𝑐𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)))
4 vex 3472 . . . . . . . . . . . . 13 𝑏 ∈ V
5 vex 3472 . . . . . . . . . . . . 13 𝑐 ∈ V
64, 5brcnv 5730 . . . . . . . . . . . 12 (𝑏𝑅𝑐𝑐𝑅𝑏)
7 df-br 5043 . . . . . . . . . . . 12 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
85, 4brcnv 5730 . . . . . . . . . . . 12 (𝑐𝑅𝑏𝑏𝑅𝑐)
96, 7, 83bitr3ri 305 . . . . . . . . . . 11 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
10 vex 3472 . . . . . . . . . . . . 13 𝑎 ∈ V
114, 10brcnv 5730 . . . . . . . . . . . 12 (𝑏𝑅𝑎𝑎𝑅𝑏)
12 df-br 5043 . . . . . . . . . . . 12 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
1310, 4brcnv 5730 . . . . . . . . . . . 12 (𝑎𝑅𝑏𝑏𝑅𝑎)
1411, 12, 133bitr3ri 305 . . . . . . . . . . 11 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
159, 14anbi12ci 630 . . . . . . . . . 10 ((𝑏𝑅𝑐𝑏𝑅𝑎) ↔ (⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅))
1615imbi1i 353 . . . . . . . . 9 (((𝑏𝑅𝑐𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ ((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
173, 16bitr3i 280 . . . . . . . 8 ((𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
1817albii 1821 . . . . . . 7 (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
192, 18bitr3i 280 . . . . . 6 ((𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
2019albii 1821 . . . . 5 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑐𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
21 alcom 2163 . . . . 5 (∀𝑐𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐) ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
2220, 21bitri 278 . . . 4 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
23 opeq2 4778 . . . . . 6 (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
2423eleq1d 2898 . . . . 5 (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ 𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅))
2524mo4 2649 . . . 4 (∃*𝑎𝑏, 𝑎⟩ ∈ 𝑅 ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
26 df-mo 2622 . . . 4 (∃*𝑎𝑏, 𝑎⟩ ∈ 𝑅 ↔ ∃𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
2722, 25, 263bitr2i 302 . . 3 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∃𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
2827albii 1821 . 2 (∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
29 relcnv 5945 . . . 4 Rel 𝑅
3029biantrur 534 . . 3 (∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐) ↔ (Rel 𝑅 ∧ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐)))
31 dffun5 6347 . . 3 (Fun 𝑅 ↔ (Rel 𝑅 ∧ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐)))
3230, 31bitr4i 281 . 2 (∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐) ↔ Fun 𝑅)
331, 28, 323bitri 300 1 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  wcel 2114  ∃*wmo 2620  cop 4545   class class class wbr 5042  ccnv 5531  Rel wrel 5537  Fun wfun 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-fun 6336
This theorem is referenced by:  frege116  40611
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