Theorem predres 6360

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 4080	. . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴
2		sseqin2 4223	. . . . . 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 4320	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
5	3, 4	eqtr2i 2766	. . . 4 ⊢ ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
6		iniseg 6115	. . . . . 6 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
7	6	ineq2d 4220	. . . . 5 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}))
8		incom 4209	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
9	7, 8	eqtrdi 2793	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴))
10		iniseg 6115	. . . . . 6 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋})
11		brres 6004	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑦(𝑅 ↾ 𝐴)𝑋 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)))
12	11	abbidv 2808	. . . . . . 7 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)})
13		df-rab 3437	. . . . . . 7 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)}
14	12, 13	eqtr4di 2795	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
15	10, 14	eqtrd 2777	. . . . 5 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
16	15	ineq2d 4220	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
17	5, 9, 16	3eqtr4a 2803	. . 3 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})))
18		df-pred 6321	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
19		df-pred 6321	. . 3 ⊢ Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋}))
20	17, 18, 19	3eqtr4g 2802	. 2 ⊢ (𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
21		predprc 6359	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
22		predprc 6359	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = ∅)
23	21, 22	eqtr4d 2780	. 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 1540   ∈ wcel 2108  {cab 2714  {crab 3436  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688  Predcpred 6320

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321

This theorem is referenced by:  frmin  9789  frrlem16  9798  frr1  9799