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Theorem predeq123 5989
 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 1117 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝐴 = 𝐵)
2 cnveq 5595 . . . . 5 (𝑅 = 𝑆𝑅 = 𝑆)
323ad2ant1 1113 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
4 sneq 4452 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
543ad2ant3 1115 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → {𝑋} = {𝑌})
63, 5imaeq12d 5773 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝑅 “ {𝑋}) = (𝑆 “ {𝑌}))
71, 6ineq12d 4079 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑆 “ {𝑌})))
8 df-pred 5988 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
9 df-pred 5988 . 2 Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (𝑆 “ {𝑌}))
107, 8, 93eqtr4g 2839 1 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1068   = wceq 1507   ∩ cin 3830  {csn 4442  ◡ccnv 5407   “ cima 5411  Predcpred 5987 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-xp 5414  df-cnv 5416  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988 This theorem is referenced by:  predeq1  5990  predeq2  5991  predeq3  5992  wsuceq123  32622  wlimeq12  32627  frecseq123  32640
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