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

Theorem predeq123 5823
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 1129 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝐴 = 𝐵)
2 cnveq 5433 . . . . 5 (𝑅 = 𝑆𝑅 = 𝑆)
323ad2ant1 1125 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → 𝑅 = 𝑆)
4 sneq 4323 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
543ad2ant3 1127 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → {𝑋} = {𝑌})
63, 5imaeq12d 5607 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝑅 “ {𝑋}) = (𝑆 “ {𝑌}))
71, 6ineq12d 3963 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑆 “ {𝑌})))
8 df-pred 5822 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
9 df-pred 5822 . 2 Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (𝑆 “ {𝑌}))
107, 8, 93eqtr4g 2828 1 ((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1069   = wceq 1629  cin 3719  {csn 4313  ccnv 5247  cima 5251  Predcpred 5821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-rab 3068  df-v 3350  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4784  df-opab 4844  df-xp 5254  df-cnv 5256  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822
This theorem is referenced by:  predeq1  5824  predeq2  5825  predeq3  5826  wsuceq123  32097  wlimeq12  32102  frecseq123  32115
  Copyright terms: Public domain W3C validator