Theorem predres 6326

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 4033	. . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴
2		sseqin2 4175	. . . . . 6 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
3	1, 2	mpbi 232	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
4		dfrab2 4272	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
5	3, 4	eqtr2i 2786	. . . 4 ⊢ ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
6		iniseg 6086	. . . . . 6 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
7	6	ineq2d 4172	. . . . 5 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}))
8		incom 4161	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
9	7, 8	eqtrdi 2813	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴))
10		iniseg 6086	. . . . . 6 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋})
11		brres 5972	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑦(𝑅 ↾ 𝐴)𝑋 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)))
12	11	abbidv 2828	. . . . . . 7 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)})
13		df-rab 3415	. . . . . . 7 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)}
14	12, 13	eqtr4di 2815	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
15	10, 14	eqtrd 2797	. . . . 5 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
16	15	ineq2d 4172	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
17	5, 9, 16	3eqtr4a 2823	. . 3 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})))
18		df-pred 6288	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
19		df-pred 6288	. . 3 ⊢ Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋}))
20	17, 18, 19	3eqtr4g 2822	. 2 ⊢ (𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
21		predprc 6325	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
22		predprc 6325	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = ∅)
23	21, 22	eqtr4d 2800	. 2 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
24	20, 23	pm2.61i 183	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {cab 2740  {crab 3414  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582   class class class wbr 5100  ^◡ccnv 5646   ↾ cres 5649   “ cima 5650  Predcpred 6287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288

This theorem is referenced by:  frmin  9707  frrlem16  9716  frr1  9717