Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege123 Structured version   Visualization version   GIF version

Theorem frege123 38780
Description: Lemma for frege124 38781. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege123.x 𝑋𝑈
frege123.y 𝑌𝑉
Assertion
Ref Expression
frege123 ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))))
Distinct variable groups:   𝑅,𝑎   𝑋,𝑎   𝑌,𝑎
Allowed substitution hints:   𝑈(𝑎)   𝑀(𝑎)   𝑉(𝑎)

Proof of Theorem frege123
StepHypRef Expression
1 frege123.x . . . 4 𝑋𝑈
2 frege123.y . . . 4 𝑌𝑉
3 vex 3394 . . . 4 𝑎 ∈ V
41, 2, 3frege122 38779 . . 3 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎)))
54alrimdv 2020 . 2 (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎)))
6 frege19 38618 . 2 ((Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎))) → ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)))))
75, 6ax-mp 5 1 ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635  wcel 2156  Vcvv 3391  cun 3767   class class class wbr 4844   I cid 5218  ccnv 5310  Fun wfun 6095  cfv 6101  t+ctcl 13949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-frege1 38584  ax-frege2 38585  ax-frege8 38603  ax-frege52a 38651  ax-frege58b 38695
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ifp 1079  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-fun 6103
This theorem is referenced by:  frege124  38781
  Copyright terms: Public domain W3C validator