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Theorem csbpredg 6260
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 4400 . . 3 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋}))
2 csbima12 6032 . . . . 5 𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
3 csbcnv 5840 . . . . . . 7 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
43imaeq1i 6011 . . . . . 6 (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
5 csbsng 4670 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑋} = {𝐴 / 𝑥𝑋})
65imaeq2d 6014 . . . . . 6 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
74, 6eqtr3id 2787 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
82, 7eqtrid 2785 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
98ineq2d 4173 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
101, 9eqtrid 2785 . 2 (𝐴𝑉𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
11 df-pred 6254 . . 3 Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (𝑅 “ {𝑋}))
1211csbeq2i 3864 . 2 𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋}))
13 df-pred 6254 . 2 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
1410, 12, 133eqtr4g 2798 1 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  csb 3856  cin 3910  {csn 4587  ccnv 5633  cima 5637  Predcpred 6253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254
This theorem is referenced by:  csbfrecsg  8216
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