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Theorem predeq123 6295
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 1134 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝐴 = 𝐵)
2 cnveq 5867 . . . . 5 (𝑅 = 𝑆𝑅 = 𝑆)
323ad2ant1 1130 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
4 sneq 4633 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
543ad2ant3 1132 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → {𝑋} = {𝑌})
63, 5imaeq12d 6054 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝑅 “ {𝑋}) = (𝑆 “ {𝑌}))
71, 6ineq12d 4208 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑆 “ {𝑌})))
8 df-pred 6294 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
9 df-pred 6294 . 2 Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (𝑆 “ {𝑌}))
107, 8, 93eqtr4g 2791 1 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  cin 3942  {csn 4623  ccnv 5668  cima 5672  Predcpred 6293
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-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294
This theorem is referenced by:  predeq1  6296  predeq2  6297  predeq3  6298  frecseq123  8268  wsuceq123  35319  wlimeq12  35324
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