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Theorem trpred 6352
Description: The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.)
Assertion
Ref Expression
trpred ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)

Proof of Theorem trpred
StepHypRef Expression
1 predep 6351 . . 3 (𝑋𝐴 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
21adantl 481 . 2 ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
3 trss 5270 . . . 4 (Tr 𝐴 → (𝑋𝐴𝑋𝐴))
43imp 406 . . 3 ((Tr 𝐴𝑋𝐴) → 𝑋𝐴)
5 sseqin2 4223 . . 3 (𝑋𝐴 ↔ (𝐴𝑋) = 𝑋)
64, 5sylib 218 . 2 ((Tr 𝐴𝑋𝐴) → (𝐴𝑋) = 𝑋)
72, 6eqtrd 2777 1 ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cin 3950  wss 3951  Tr wtr 5259   E cep 5583  Predcpred 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321
This theorem is referenced by:  predon  7806  omsinds  7908  trfr  44979
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