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Theorem csbpredg 33302
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 4043 . . 3 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋}))
2 csbima12 5518 . . . . 5 𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
3 csbcnv 5338 . . . . . . 7 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
43imaeq1i 5498 . . . . . 6 (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋})
5 csbsng 4275 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑋} = {𝐴 / 𝑥𝑋})
65imaeq2d 5501 . . . . . 6 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
74, 6syl5eqr 2699 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥{𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
82, 7syl5eq 2697 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝑅 “ {𝑋}) = (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
98ineq2d 3847 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐷𝐴 / 𝑥(𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
101, 9syl5eq 2697 . 2 (𝐴𝑉𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋})) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋})))
11 df-pred 5718 . . 3 Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (𝑅 “ {𝑋}))
1211csbeq2i 4026 . 2 𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = 𝐴 / 𝑥(𝐷 ∩ (𝑅 “ {𝑋}))
13 df-pred 5718 . 2 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋) = (𝐴 / 𝑥𝐷 ∩ (𝐴 / 𝑥𝑅 “ {𝐴 / 𝑥𝑋}))
1410, 12, 133eqtr4g 2710 1 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  csb 3566  cin 3606  {csn 4210  ccnv 5142  cima 5146  Predcpred 5717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718
This theorem is referenced by:  csbwrecsg  33303
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