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Theorem predeq123 6323
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
predeq123 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))

Proof of Theorem predeq123
StepHypRef Expression
1 simp2 1136 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝐴 = 𝐵)
2 cnveq 5886 . . . . 5 (𝑅 = 𝑆𝑅 = 𝑆)
323ad2ant1 1132 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
4 sneq 4640 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
543ad2ant3 1134 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → {𝑋} = {𝑌})
63, 5imaeq12d 6080 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝑅 “ {𝑋}) = (𝑆 “ {𝑌}))
71, 6ineq12d 4228 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑆 “ {𝑌})))
8 df-pred 6322 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
9 df-pred 6322 . 2 Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (𝑆 “ {𝑌}))
107, 8, 93eqtr4g 2799 1 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  cin 3961  {csn 4630  ccnv 5687  cima 5691  Predcpred 6321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322
This theorem is referenced by:  predeq1  6324  predeq2  6325  predeq3  6326  frecseq123  8305  wsuceq123  35795  wlimeq12  35800
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