Theorem predtrss 6316

Description: If 𝑅 is transitive over 𝐴 and 𝑌𝑅𝑋, then Pred(𝑅, 𝐴, 𝑌) is a subclass of Pred(𝑅, 𝐴, 𝑋). (Contributed by Scott Fenton, 28-Oct-2024.)

Assertion

Ref	Expression
predtrss	⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))

Proof of Theorem predtrss

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2		predel 6315	. . . . . . . 8 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴)
3	2	3ad2ant2 1134	. . . . . . 7 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → 𝑌 ∈ 𝐴)
4	3	adantr 480	. . . . . 6 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌 ∈ 𝐴)
5		brxp 5708	. . . . . 6 ⊢ (𝑧(𝐴 × 𝐴)𝑌 ↔ (𝑧 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
6	1, 4, 5	sylanbrc 583	. . . . 5 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧(𝐴 × 𝐴)𝑌)
7		brin 5176	. . . . . 6 ⊢ (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ↔ (𝑧𝑅𝑌 ∧ 𝑧(𝐴 × 𝐴)𝑌))
8		elpredimg 6310	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
9	8	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → 𝑌𝑅𝑋)
10	9	3adant1 1130	. . . . . . . . 9 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → 𝑌𝑅𝑋)
11	10	adantr 480	. . . . . . . 8 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌𝑅𝑋)
12		simpl3 1194	. . . . . . . . 9 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑋 ∈ 𝐴)
13		brxp 5708	. . . . . . . . 9 ⊢ (𝑌(𝐴 × 𝐴)𝑋 ↔ (𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴))
14	4, 12, 13	sylanbrc 583	. . . . . . . 8 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌(𝐴 × 𝐴)𝑋)
15		brin 5176	. . . . . . . 8 ⊢ (𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋 ↔ (𝑌𝑅𝑋 ∧ 𝑌(𝐴 × 𝐴)𝑋))
16	11, 14, 15	sylanbrc 583	. . . . . . 7 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋)
17		breq2 5128	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌))
18		breq1 5127	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋 ↔ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋))
19	17, 18	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋) ↔ (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ∧ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
20	19	spcegv 3581	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ∧ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
21	20	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ∧ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
22	21	adantr 480	. . . . . . . . 9 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ∧ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
23		vex 3468	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
24		brcog 5851	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ 𝑋 ∈ 𝐴) → (𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋 ↔ ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
25	23, 12, 24	sylancr 587	. . . . . . . . 9 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋 ↔ ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
26	22, 25	sylibrd 259	. . . . . . . 8 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ∧ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → 𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋))
27		simpl1 1192	. . . . . . . . 9 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)
28	27	ssbrd 5167	. . . . . . . 8 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋 → 𝑧𝑅𝑋))
29	26, 28	syld 47	. . . . . . 7 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ∧ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → 𝑧𝑅𝑋))
30	16, 29	mpan2d 694	. . . . . 6 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 → 𝑧𝑅𝑋))
31	7, 30	biimtrrid 243	. . . . 5 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑧𝑅𝑌 ∧ 𝑧(𝐴 × 𝐴)𝑌) → 𝑧𝑅𝑋))
32	6, 31	mpan2d 694	. . . 4 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑧𝑅𝑌 → 𝑧𝑅𝑋))
33	32	imdistanda 571	. . 3 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑌) → (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑋)))
34	23	elpred 6312	. . . 4 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑌)))
35	34	3ad2ant2 1134	. . 3 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑌)))
36	23	elpred 6312	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑋)))
37	36	3ad2ant3 1135	. . 3 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑋)))
38	33, 35, 37	3imtr4d 294	. 2 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) → 𝑧 ∈ Pred(𝑅, 𝐴, 𝑋)))
39	38	ssrdv 3969	1 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931   class class class wbr 5124   × cxp 5657   ∘ ccom 5663  Predcpred 6294

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295

This theorem is referenced by:  predpo  6317  frmin  9768  frrlem16  9777