Theorem csbpredg 6259

Description: Move class substitution in and out of the predecessor class of a relation. (Contributed by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csbpredg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋))

Proof of Theorem csbpredg

Step	Hyp	Ref	Expression
1		csbin 4395	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}))
2		csbima12 6034	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋})
3		csbcnv 5830	. . . . . . 7 ⊢ ◡⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌◡𝑅
4	3	imaeq1i 6012	. . . . . 6 ⊢ (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋})
5		csbsng 4662	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑋} = {⦋𝐴 / 𝑥⦌𝑋})
6	5	imaeq2d 6015	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
7	4, 6	eqtr3id 2778	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
8	2, 7	eqtrid 2776	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
9	8	ineq2d 4173	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})))
10	1, 9	eqtrid 2776	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})))
11		df-pred 6253	. . 3 ⊢ Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (◡𝑅 “ {𝑋}))
12	11	csbeq2i 3861	. 2 ⊢ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋}))
13		df-pred 6253	. 2 ⊢ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
14	10, 12, 13	3eqtr4g 2789	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⦋csb 3853   ∩ cin 3904  {csn 4579  ^◡ccnv 5622   “ cima 5626  Predcpred 6252

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253

This theorem is referenced by:  csbfrecsg  8224