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Theorem dffrege115 42729
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 2157 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
2 19.21v 1943 . . . . . . 7 (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
3 impexp 452 . . . . . . . . 9 (((𝑏𝑅𝑐𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)))
4 vex 3479 . . . . . . . . . . . . 13 𝑏 ∈ V
5 vex 3479 . . . . . . . . . . . . 13 𝑐 ∈ V
64, 5brcnv 5883 . . . . . . . . . . . 12 (𝑏𝑅𝑐𝑐𝑅𝑏)
7 df-br 5150 . . . . . . . . . . . 12 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
85, 4brcnv 5883 . . . . . . . . . . . 12 (𝑐𝑅𝑏𝑏𝑅𝑐)
96, 7, 83bitr3ri 302 . . . . . . . . . . 11 (𝑏𝑅𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅)
10 vex 3479 . . . . . . . . . . . . 13 𝑎 ∈ V
114, 10brcnv 5883 . . . . . . . . . . . 12 (𝑏𝑅𝑎𝑎𝑅𝑏)
12 df-br 5150 . . . . . . . . . . . 12 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
1310, 4brcnv 5883 . . . . . . . . . . . 12 (𝑎𝑅𝑏𝑏𝑅𝑎)
1411, 12, 133bitr3ri 302 . . . . . . . . . . 11 (𝑏𝑅𝑎 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝑅)
159, 14anbi12ci 629 . . . . . . . . . 10 ((𝑏𝑅𝑐𝑏𝑅𝑎) ↔ (⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅))
1615imbi1i 350 . . . . . . . . 9 (((𝑏𝑅𝑐𝑏𝑅𝑎) → 𝑎 = 𝑐) ↔ ((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
173, 16bitr3i 277 . . . . . . . 8 ((𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
1817albii 1822 . . . . . . 7 (∀𝑎(𝑏𝑅𝑐 → (𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
192, 18bitr3i 277 . . . . . 6 ((𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
2019albii 1822 . . . . 5 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑐𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
21 alcom 2157 . . . . 5 (∀𝑐𝑎((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐) ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
2220, 21bitri 275 . . . 4 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
23 opeq2 4875 . . . . . 6 (𝑎 = 𝑐 → ⟨𝑏, 𝑎⟩ = ⟨𝑏, 𝑐⟩)
2423eleq1d 2819 . . . . 5 (𝑎 = 𝑐 → (⟨𝑏, 𝑎⟩ ∈ 𝑅 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑅))
2524mo4 2561 . . . 4 (∃*𝑎𝑏, 𝑎⟩ ∈ 𝑅 ↔ ∀𝑎𝑐((⟨𝑏, 𝑎⟩ ∈ 𝑅 ∧ ⟨𝑏, 𝑐⟩ ∈ 𝑅) → 𝑎 = 𝑐))
26 df-mo 2535 . . . 4 (∃*𝑎𝑏, 𝑎⟩ ∈ 𝑅 ↔ ∃𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
2722, 25, 263bitr2i 299 . . 3 (∀𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∃𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
2827albii 1822 . 2 (∀𝑏𝑐(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐))
29 relcnv 6104 . . . 4 Rel 𝑅
3029biantrur 532 . . 3 (∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐) ↔ (Rel 𝑅 ∧ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐)))
31 dffun5 6561 . . 3 (Fun 𝑅 ↔ (Rel 𝑅 ∧ ∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐)))
3230, 31bitr4i 278 . 2 (∀𝑏𝑐𝑎(⟨𝑏, 𝑎⟩ ∈ 𝑅𝑎 = 𝑐) ↔ Fun 𝑅)
331, 28, 323bitri 297 1 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  ∃*wmo 2533  cop 4635   class class class wbr 5149  ccnv 5676  Rel wrel 5682  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546
This theorem is referenced by:  frege116  42730
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