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Theorem csbpredg 6329
Description: Move class substitution in and out of the predecessor class of a relation. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbpredg (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))

Proof of Theorem csbpredg
StepHypRef Expression
1 csbin 4448 . . 3 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋}))
2 csbima12 6099 . . . . 5 𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
3 csbcnv 5897 . . . . . . 7 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
43imaeq1i 6077 . . . . . 6 (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
5 csbsng 4713 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑋} = {𝐴 / 𝑥𝑋})
65imaeq2d 6080 . . . . . 6 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
74, 6eqtr3id 2789 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
82, 7eqtrid 2787 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
98ineq2d 4228 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
101, 9eqtrid 2787 . 2 (𝐴𝑉𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
11 df-pred 6323 . . 3 Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (𝑅 “ {𝑋}))
1211csbeq2i 3916 . 2 𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋}))
13 df-pred 6323 . 2 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
1410, 12, 133eqtr4g 2800 1 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  csb 3908  cin 3962  {csn 4631  ccnv 5688  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-nfc 2890  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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:  csbfrecsg  8308
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