Theorem predres 6337

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 4028	. . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴
2		sseqin2 4169	. . . . . 6 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
3	1, 2	mpbi 233	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
4		dfrab2 4266	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
5	3, 4	eqtr2i 2784	. . . 4 ⊢ ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
6		iniseg 6093	. . . . . 6 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
7	6	ineq2d 4166	. . . . 5 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}))
8		incom 4155	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
9	7, 8	eqtrdi 2811	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴))
10		iniseg 6093	. . . . . 6 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋})
11		brres 5979	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑦(𝑅 ↾ 𝐴)𝑋 ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)))
12	11	abbidv 2826	. . . . . . 7 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)})
13		df-rab 3413	. . . . . . 7 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)}
14	12, 13	eqtr4di 2813	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
15	10, 14	eqtrd 2795	. . . . 5 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
16	15	ineq2d 4166	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
17	5, 9, 16	3eqtr4a 2821	. . 3 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})))
18		df-pred 6299	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
19		df-pred 6299	. . 3 ⊢ Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋}))
20	17, 18, 19	3eqtr4g 2820	. 2 ⊢ (𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
21		predprc 6336	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
22		predprc 6336	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = ∅)
23	21, 22	eqtr4d 2798	. 2 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
24	20, 23	pm2.61i 184	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  {crab 3412  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584   class class class wbr 5103  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658  Predcpred 6298

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299

This theorem is used by:  frmin  9731  frrlem16  9740  frr1  9741