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Theorem predres 6360
Description: Predecessor class is unaffected by restriction to the base class. (Contributed by Scott Fenton, 25-Nov-2024.)
Assertion
Ref Expression
predres Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅𝐴), 𝐴, 𝑋)

Proof of Theorem predres
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 4080 . . . . . 6 {𝑦𝐴𝑦𝑅𝑋} ⊆ 𝐴
2 sseqin2 4223 . . . . . 6 ({𝑦𝐴𝑦𝑅𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋}) = {𝑦𝐴𝑦𝑅𝑋})
31, 2mpbi 230 . . . . 5 (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋}) = {𝑦𝐴𝑦𝑅𝑋}
4 dfrab2 4320 . . . . 5 {𝑦𝐴𝑦𝑅𝑋} = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
53, 4eqtr2i 2766 . . . 4 ({𝑦𝑦𝑅𝑋} ∩ 𝐴) = (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋})
6 iniseg 6115 . . . . . 6 (𝑋 ∈ V → (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋})
76ineq2d 4220 . . . . 5 (𝑋 ∈ V → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦𝑦𝑅𝑋}))
8 incom 4209 . . . . 5 (𝐴 ∩ {𝑦𝑦𝑅𝑋}) = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
97, 8eqtrdi 2793 . . . 4 (𝑋 ∈ V → (𝐴 ∩ (𝑅 “ {𝑋})) = ({𝑦𝑦𝑅𝑋} ∩ 𝐴))
10 iniseg 6115 . . . . . 6 (𝑋 ∈ V → ((𝑅𝐴) “ {𝑋}) = {𝑦𝑦(𝑅𝐴)𝑋})
11 brres 6004 . . . . . . . 8 (𝑋 ∈ V → (𝑦(𝑅𝐴)𝑋 ↔ (𝑦𝐴𝑦𝑅𝑋)))
1211abbidv 2808 . . . . . . 7 (𝑋 ∈ V → {𝑦𝑦(𝑅𝐴)𝑋} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑋)})
13 df-rab 3437 . . . . . . 7 {𝑦𝐴𝑦𝑅𝑋} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑋)}
1412, 13eqtr4di 2795 . . . . . 6 (𝑋 ∈ V → {𝑦𝑦(𝑅𝐴)𝑋} = {𝑦𝐴𝑦𝑅𝑋})
1510, 14eqtrd 2777 . . . . 5 (𝑋 ∈ V → ((𝑅𝐴) “ {𝑋}) = {𝑦𝐴𝑦𝑅𝑋})
1615ineq2d 4220 . . . 4 (𝑋 ∈ V → (𝐴 ∩ ((𝑅𝐴) “ {𝑋})) = (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋}))
175, 9, 163eqtr4a 2803 . . 3 (𝑋 ∈ V → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ ((𝑅𝐴) “ {𝑋})))
18 df-pred 6321 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
19 df-pred 6321 . . 3 Pred((𝑅𝐴), 𝐴, 𝑋) = (𝐴 ∩ ((𝑅𝐴) “ {𝑋}))
2017, 18, 193eqtr4g 2802 . 2 (𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅𝐴), 𝐴, 𝑋))
21 predprc 6359 . . 3 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
22 predprc 6359 . . 3 𝑋 ∈ V → Pred((𝑅𝐴), 𝐴, 𝑋) = ∅)
2321, 22eqtr4d 2780 . 2 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅𝐴), 𝐴, 𝑋))
2420, 23pm2.61i 182 1 Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅𝐴), 𝐴, 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wcel 2108  {cab 2714  {crab 3436  Vcvv 3480  cin 3950  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  ccnv 5684  cres 5687  cima 5688  Predcpred 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321
This theorem is referenced by:  frmin  9789  frrlem16  9798  frr1  9799
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