Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege116 Structured version   Visualization version   GIF version

Theorem frege116 41587
Description: One direction of dffrege115 41586. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege116.x 𝑋𝑈
Assertion
Ref Expression
frege116 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
Distinct variable groups:   𝑎,𝑏,𝑅   𝑋,𝑎,𝑏
Allowed substitution hints:   𝑈(𝑎,𝑏)

Proof of Theorem frege116
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 dffrege115 41586 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
2 frege116.x . . . 4 𝑋𝑈
32frege68c 41539 . . 3 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅[𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
4 sbcal 3780 . . . 4 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
5 sbcimg 3767 . . . . . . 7 (𝑋𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
62, 5ax-mp 5 . . . . . 6 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
7 sbcbr2g 5132 . . . . . . . . 9 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐))
82, 7ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐)
9 csbvarg 4365 . . . . . . . . . 10 (𝑋𝑈𝑋 / 𝑐𝑐 = 𝑋)
102, 9ax-mp 5 . . . . . . . . 9 𝑋 / 𝑐𝑐 = 𝑋
1110breq2i 5082 . . . . . . . 8 (𝑏𝑅𝑋 / 𝑐𝑐𝑏𝑅𝑋)
128, 11bitri 274 . . . . . . 7 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋)
13 sbcal 3780 . . . . . . . 8 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐))
14 sbcimg 3767 . . . . . . . . . . 11 (𝑋𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐)))
152, 14ax-mp 5 . . . . . . . . . 10 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐))
16 sbcg 3795 . . . . . . . . . . . 12 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎))
172, 16ax-mp 5 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎)
18 sbceq2g 4350 . . . . . . . . . . . . 13 (𝑋𝑈 → ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐))
192, 18ax-mp 5 . . . . . . . . . . . 12 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐)
2010eqeq2i 2751 . . . . . . . . . . . 12 (𝑎 = 𝑋 / 𝑐𝑐𝑎 = 𝑋)
2119, 20bitri 274 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋)
2217, 21imbi12i 351 . . . . . . . . . 10 (([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2315, 22bitri 274 . . . . . . . . 9 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2423albii 1822 . . . . . . . 8 (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2513, 24bitri 274 . . . . . . 7 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2612, 25imbi12i 351 . . . . . 6 (([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
276, 26bitri 274 . . . . 5 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
2827albii 1822 . . . 4 (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
294, 28bitri 274 . . 3 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
303, 29syl6ib 250 . 2 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))))
311, 30ax-mp 5 1 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  wcel 2106  [wsbc 3716  csb 3832   class class class wbr 5074  ccnv 5588  Fun wfun 6427
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-frege1 41398  ax-frege2 41399  ax-frege8 41417  ax-frege52a 41465  ax-frege58b 41509
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-fun 6435
This theorem is referenced by:  frege117  41588
  Copyright terms: Public domain W3C validator