Theorem csbpredg 6310

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 4408	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}))
2		csbima12 6083	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋})
3		csbcnv 5874	. . . . . . 7 ⊢ ◡⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌◡𝑅
4	3	imaeq1i 6061	. . . . . 6 ⊢ (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋})
5		csbsng 4675	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑋} = {⦋𝐴 / 𝑥⦌𝑋})
6	5	imaeq2d 6064	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
7	4, 6	eqtr3id 2812	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
8	2, 7	eqtrid 2810	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))
9	8	ineq2d 4174	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})))
10	1, 9	eqtrid 2810	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})))
11		df-pred 6304	. . 3 ⊢ Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (◡𝑅 “ {𝑋}))
12	11	csbeq2i 3862	. 2 ⊢ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋}))
13		df-pred 6304	. 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 2823	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ⦋csb 3854   ∩ cin 3905  {csn 4590  ^◡ccnv 5662   “ cima 5666  Predcpred 6303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304

This theorem is referenced by:  csbfrecsg  8282