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Theorem csbpredg 33606
Description: Move class substitution in and out of the predecessor class of a relationship. (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 4172 . . 3 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋}))
2 csbima12 5665 . . . . 5 𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
3 csbcnv 5474 . . . . . . 7 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
43imaeq1i 5645 . . . . . 6 (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
5 csbsng 4399 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑋} = {𝐴 / 𝑥𝑋})
65imaeq2d 5648 . . . . . 6 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
74, 6syl5eqr 2813 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
82, 7syl5eq 2811 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
98ineq2d 3976 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
101, 9syl5eq 2811 . 2 (𝐴𝑉𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
11 df-pred 5865 . . 3 Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (𝑅 “ {𝑋}))
1211csbeq2i 4154 . 2 𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋}))
13 df-pred 5865 . 2 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
1410, 12, 133eqtr4g 2824 1 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  csb 3691  cin 3731  {csn 4334  ccnv 5276  cima 5280  Predcpred 5864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865
This theorem is referenced by:  csbwrecsg  33607
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