Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege121 Structured version   Visualization version   GIF version

Theorem frege121 40337
 Description: Lemma for frege122 40338. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege116.x 𝑋𝑈
frege118.y 𝑌𝑉
frege120.a 𝐴𝑊
Assertion
Ref Expression
frege121 ((𝐴 = 𝑋𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴))))

Proof of Theorem frege121
StepHypRef Expression
1 frege116.x . . 3 𝑋𝑈
2 frege118.y . . 3 𝑌𝑉
3 frege120.a . . 3 𝐴𝑊
41, 2, 3frege120 40336 . 2 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))
5 frege20 40181 . 2 ((Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋))) → ((𝐴 = 𝑋𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴)))))
64, 5ax-mp 5 1 ((𝐴 = 𝑋𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∪ cun 3936   class class class wbr 5068   I cid 5461  ◡ccnv 5556  Fun wfun 6351  ‘cfv 6357  t+ctcl 14347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-frege1 40143  ax-frege2 40144  ax-frege8 40162  ax-frege52a 40210  ax-frege58b 40254 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-fun 6359 This theorem is referenced by:  frege122  40338
 Copyright terms: Public domain W3C validator