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Theorem frege122 43554
Description: If 𝑋 is a result of an application of the single-valued procedure 𝑅 to 𝑌, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑋. Proposition 122 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
frege122 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴)))

Proof of Theorem frege122
StepHypRef Expression
1 frege120.a . . 3 𝐴𝑊
21frege112 43544 . 2 (𝐴 = 𝑋𝑋((t+‘𝑅) ∪ I )𝐴)
3 frege116.x . . 3 𝑋𝑈
4 frege118.y . . 3 𝑌𝑉
53, 4, 1frege121 43553 . 2 ((𝐴 = 𝑋𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴))))
62, 5ax-mp 5 1 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cun 3942   class class class wbr 5149   I cid 5575  ccnv 5677  Fun wfun 6543  cfv 6549  t+ctcl 14968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-frege1 43359  ax-frege2 43360  ax-frege8 43378  ax-frege52a 43426  ax-frege58b 43470
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-fun 6551
This theorem is referenced by:  frege123  43555
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