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Theorem csbpredg 6327
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 4442 . . 3 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋}))
2 csbima12 6097 . . . . 5 𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
3 csbcnv 5894 . . . . . . 7 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
43imaeq1i 6075 . . . . . 6 (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
5 csbsng 4708 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑋} = {𝐴 / 𝑥𝑋})
65imaeq2d 6078 . . . . . 6 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
74, 6eqtr3id 2791 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
82, 7eqtrid 2789 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
98ineq2d 4220 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
101, 9eqtrid 2789 . 2 (𝐴𝑉𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
11 df-pred 6321 . . 3 Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (𝑅 “ {𝑋}))
1211csbeq2i 3907 . 2 𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋}))
13 df-pred 6321 . 2 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
1410, 12, 133eqtr4g 2802 1 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  csb 3899  cin 3950  {csn 4626  ccnv 5684  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-nfc 2892  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  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:  csbfrecsg  8309
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