Theorem predtrss 6325

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 490	. . . . . 6 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
2		predel 6324	. . . . . . . 8 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴)
3	2	3ad2ant2 1152	. . . . . . 7 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → 𝑌 ∈ 𝐴)
4	3	adantr 486	. . . . . 6 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌 ∈ 𝐴)
5		brxp 5699	. . . . . 6 ⊢ (𝑧(𝐴 × 𝐴)𝑌 ↔ (𝑧 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
6	1, 4, 5	sylanbrc 595	. . . . 5 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧(𝐴 × 𝐴)𝑌)
7		brin 5156	. . . . . 6 ⊢ (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ↔ (𝑧𝑅𝑌 ∧ 𝑧(𝐴 × 𝐴)𝑌))
8		elpredimg 6319	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
9	8	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → 𝑌𝑅𝑋)
10	9	3adant1 1148	. . . . . . . . 9 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → 𝑌𝑅𝑋)
11	10	adantr 486	. . . . . . . 8 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌𝑅𝑋)
12		simpl3 1212	. . . . . . . . 9 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑋 ∈ 𝐴)
13		brxp 5699	. . . . . . . . 9 ⊢ (𝑌(𝐴 × 𝐴)𝑋 ↔ (𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴))
14	4, 12, 13	sylanbrc 595	. . . . . . . 8 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌(𝐴 × 𝐴)𝑋)
15		brin 5156	. . . . . . . 8 ⊢ (𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋 ↔ (𝑌𝑅𝑋 ∧ 𝑌(𝐴 × 𝐴)𝑋))
16	11, 14, 15	sylanbrc 595	. . . . . . 7 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋)
17		breq2 5106	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌))
18		breq1 5105	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋 ↔ 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋))
19	17, 18	anbi12d 644	. . . . . . . . . . . 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 3551	. . . . . . . . . . 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 1152	. . . . . . . . . 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 486	. . . . . . . . 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 3454	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
24		brcog 5843	. . . . . . . . . 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 599	. . . . . . . . 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 262	. . . . . . . 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 1210	. . . . . . . . 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 5147	. . . . . . . 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 48	. . . . . . 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 707	. . . . . 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 246	. . . . 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 707	. . . 4 ⊢ (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑧𝑅𝑌 → 𝑧𝑅𝑋))
33	32	imdistanda 582	. . 3 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑌) → (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑋)))
34	23	elpred 6321	. . . 4 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑌)))
35	34	3ad2ant2 1152	. . 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 6321	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧𝑅𝑋)))
37	36	3ad2ant3 1153	. . 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 297	. 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 3914	1 ⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   class class class wbr 5102   × cxp 5648   ∘ ccom 5654  Predcpred 6303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304

This theorem is used by:  predpo  6326  frmin  9757  frrlem16  9766