Theorem predres 6338

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 4077	. . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴
2		sseqin2 4215	. . . . . 6 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
3	1, 2	mpbi 229	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
4		dfrab2 4310	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
5	3, 4	eqtr2i 2762	. . . 4 ⊢ ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
6		iniseg 6094	. . . . . 6 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
7	6	ineq2d 4212	. . . . 5 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}))
8		incom 4201	. . . . 5 ⊢ (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
9	7, 8	eqtrdi 2789	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴))
10		iniseg 6094	. . . . . 6 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋})
11		brres 5987	. . . . . . . 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 3434	. . . . . . 7 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)}
14	12, 13	eqtr4di 2791	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑦 ∣ 𝑦(𝑅 ↾ 𝐴)𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
15	10, 14	eqtrd 2773	. . . . 5 ⊢ (𝑋 ∈ V → (◡(𝑅 ↾ 𝐴) “ {𝑋}) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
16	15	ineq2d 4212	. . . 4 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})) = (𝐴 ∩ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
17	5, 9, 16	3eqtr4a 2799	. . 3 ⊢ (𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋})))
18		df-pred 6298	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
19		df-pred 6298	. . 3 ⊢ Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = (𝐴 ∩ (◡(𝑅 ↾ 𝐴) “ {𝑋}))
20	17, 18, 19	3eqtr4g 2798	. 2 ⊢ (𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋))
21		predprc 6337	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
22		predprc 6337	. . 3 ⊢ (¬ 𝑋 ∈ V → Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋) = ∅)
23	21, 22	eqtr4d 2776	. 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  {crab 3433  Vcvv 3475   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  {csn 4628   class class class wbr 5148  ^◡ccnv 5675   ↾ cres 5678   “ cima 5679  Predcpred 6297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298

This theorem is referenced by:  frmin  9741  frrlem16  9750  frr1  9751