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 4391	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}))
2		csbima12 6032	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋})
3		csbcnv 5827	. . . . . . 7 ⊢ ◡⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌◡𝑅
4	3	imaeq1i 6010	. . . . . 6 ⊢ (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋})
5		csbsng 4660	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑋} = {⦋𝐴 / 𝑥⦌𝑋})
6	5	imaeq2d 6013	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
7	4, 6	eqtr3id 2782	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
8	2, 7	eqtrid 2780	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
9	8	ineq2d 4169	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})))
10	1, 9	eqtrid 2780	. 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 3854	. 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 2793	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⦋csb 3846   ∩ cin 3897  {csn 4575  ^◡ccnv 5618   “ cima 5622  Predcpred 6252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253

This theorem is referenced by:  csbfrecsg  8220