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Theorem frege121 37094
Description: Lemma for frege122 37095. 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 37093 . 2 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))
5 frege20 36938 . 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 1474  wcel 1976  cun 3537   class class class wbr 4577   I cid 4938  ccnv 5027  Fun wfun 5784  cfv 5790  t+ctcl 13518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-frege1 36900  ax-frege2 36901  ax-frege8 36919  ax-frege52a 36967  ax-frege58b 37011
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-fun 5792
This theorem is referenced by:  frege122  37095
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