Theorem predres 6303

Description: Predecessor class is unaffected by restriction to the base class. (Contributed by Scott Fenton, 25-Nov-2024.)

Assertion

Ref	Expression
predres	⊢ Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)

Proof of Theorem predres

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4020	. . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴
2		sseqin2 4163	. . . . . 6 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
3	1, 2	mpbi 230	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
4		dfrab2 4260	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
5	3, 4	eqtr2i 2760	. . . 4 ⊢ ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
6		iniseg 6062	. . . . . 6 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
7	6	ineq2d 4160	. . . . 5 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}))
8		incom 4149	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
9	7, 8	eqtrdi 2787	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴))
10		iniseg 6062	. . . . . 6 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋})
11		brres 5951	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑦(𝑅 ↾ 𝐴)𝑋 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)))
12	11	abbidv 2802	. . . . . . 7 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)})
13		df-rab 3390	. . . . . . 7 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)}
14	12, 13	eqtr4di 2789	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
15	10, 14	eqtrd 2771	. . . . 5 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
16	15	ineq2d 4160	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
17	5, 9, 16	3eqtr4a 2797	. . 3 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})))
18		df-pred 6265	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
19		df-pred 6265	. . 3 ⊢ Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋}))
20	17, 18, 19	3eqtr4g 2796	. 2 ⊢ (𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
21		predprc 6302	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
22		predprc 6302	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = ∅)
23	21, 22	eqtr4d 2774	. 2 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
24	20, 23	pm2.61i 182	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2714  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  {csn 4567   class class class wbr 5085  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634  Predcpred 6264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265

This theorem is referenced by:  frmin  9673  frrlem16  9682  frr1  9683