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Theorem predres 6362
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 4090 . . . . . 6 {𝑦𝐴𝑦𝑅𝑋} ⊆ 𝐴
2 sseqin2 4231 . . . . . 6 ({𝑦𝐴𝑦𝑅𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋}) = {𝑦𝐴𝑦𝑅𝑋})
31, 2mpbi 230 . . . . 5 (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋}) = {𝑦𝐴𝑦𝑅𝑋}
4 dfrab2 4326 . . . . 5 {𝑦𝐴𝑦𝑅𝑋} = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
53, 4eqtr2i 2764 . . . 4 ({𝑦𝑦𝑅𝑋} ∩ 𝐴) = (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋})
6 iniseg 6118 . . . . . 6 (𝑋 ∈ V → (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋})
76ineq2d 4228 . . . . 5 (𝑋 ∈ V → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦𝑦𝑅𝑋}))
8 incom 4217 . . . . 5 (𝐴 ∩ {𝑦𝑦𝑅𝑋}) = ({𝑦𝑦𝑅𝑋} ∩ 𝐴)
97, 8eqtrdi 2791 . . . 4 (𝑋 ∈ V → (𝐴 ∩ (𝑅 “ {𝑋})) = ({𝑦𝑦𝑅𝑋} ∩ 𝐴))
10 iniseg 6118 . . . . . 6 (𝑋 ∈ V → ((𝑅𝐴) “ {𝑋}) = {𝑦𝑦(𝑅𝐴)𝑋})
11 brres 6007 . . . . . . . 8 (𝑋 ∈ V → (𝑦(𝑅𝐴)𝑋 ↔ (𝑦𝐴𝑦𝑅𝑋)))
1211abbidv 2806 . . . . . . 7 (𝑋 ∈ V → {𝑦𝑦(𝑅𝐴)𝑋} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑋)})
13 df-rab 3434 . . . . . . 7 {𝑦𝐴𝑦𝑅𝑋} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑋)}
1412, 13eqtr4di 2793 . . . . . 6 (𝑋 ∈ V → {𝑦𝑦(𝑅𝐴)𝑋} = {𝑦𝐴𝑦𝑅𝑋})
1510, 14eqtrd 2775 . . . . 5 (𝑋 ∈ V → ((𝑅𝐴) “ {𝑋}) = {𝑦𝐴𝑦𝑅𝑋})
1615ineq2d 4228 . . . 4 (𝑋 ∈ V → (𝐴 ∩ ((𝑅𝐴) “ {𝑋})) = (𝐴 ∩ {𝑦𝐴𝑦𝑅𝑋}))
175, 9, 163eqtr4a 2801 . . 3 (𝑋 ∈ V → (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ ((𝑅𝐴) “ {𝑋})))
18 df-pred 6323 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
19 df-pred 6323 . . 3 Pred((𝑅𝐴), 𝐴, 𝑋) = (𝐴 ∩ ((𝑅𝐴) “ {𝑋}))
2017, 18, 193eqtr4g 2800 . 2 (𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅𝐴), 𝐴, 𝑋))
21 predprc 6361 . . 3 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
22 predprc 6361 . . 3 𝑋 ∈ V → Pred((𝑅𝐴), 𝐴, 𝑋) = ∅)
2321, 22eqtr4d 2778 . 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 1537  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478  cin 3962  wss 3963  c0 4339  {csn 4631   class class class wbr 5148  ccnv 5688  cres 5691  cima 5692  Predcpred 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323
This theorem is referenced by:  frmin  9787  frrlem16  9796  frr1  9797
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