MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  predeq123 Structured version   Visualization version   GIF version

Theorem predeq123 5669
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 1060 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝐴 = 𝐵)
2 cnveq 5285 . . . . 5 (𝑅 = 𝑆𝑅 = 𝑆)
323ad2ant1 1080 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
4 sneq 4178 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
543ad2ant3 1082 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → {𝑋} = {𝑌})
63, 5imaeq12d 5455 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝑅 “ {𝑋}) = (𝑆 “ {𝑌}))
71, 6ineq12d 3807 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑆 “ {𝑌})))
8 df-pred 5668 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
9 df-pred 5668 . 2 Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (𝑆 “ {𝑌}))
107, 8, 93eqtr4g 2679 1 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  cin 3566  {csn 4168  ccnv 5103  cima 5107  Predcpred 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668
This theorem is referenced by:  predeq1  5670  predeq2  5671  predeq3  5672  wsuceq123  31734  wlimeq12  31739
  Copyright terms: Public domain W3C validator