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Theorem predtrss 6333
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))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 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.
StepHypRef Expression
1 simpr 483 . . . . . 6 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑧𝐴)
2 predel 6331 . . . . . . . 8 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
323ad2ant2 1131 . . . . . . 7 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 𝑌𝐴)
43adantr 479 . . . . . 6 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌𝐴)
5 brxp 5731 . . . . . 6 (𝑧(𝐴 × 𝐴)𝑌 ↔ (𝑧𝐴𝑌𝐴))
61, 4, 5sylanbrc 581 . . . . 5 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑧(𝐴 × 𝐴)𝑌)
7 brin 5204 . . . . . 6 (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ↔ (𝑧𝑅𝑌𝑧(𝐴 × 𝐴)𝑌))
8 predbrg 6332 . . . . . . . . . . 11 ((𝑋𝐴𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
98ancoms 457 . . . . . . . . . 10 ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 𝑌𝑅𝑋)
1093adant1 1127 . . . . . . . . 9 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 𝑌𝑅𝑋)
1110adantr 479 . . . . . . . 8 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌𝑅𝑋)
12 simpl3 1190 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑋𝐴)
13 brxp 5731 . . . . . . . . 9 (𝑌(𝐴 × 𝐴)𝑋 ↔ (𝑌𝐴𝑋𝐴))
144, 12, 13sylanbrc 581 . . . . . . . 8 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌(𝐴 × 𝐴)𝑋)
15 brin 5204 . . . . . . . 8 (𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋 ↔ (𝑌𝑅𝑋𝑌(𝐴 × 𝐴)𝑋))
1611, 14, 15sylanbrc 581 . . . . . . 7 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋)
17 breq2 5156 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌))
18 breq1 5155 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋))
1917, 18anbi12d 630 . . . . . . . . . . . 12 (𝑦 = 𝑌 → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋) ↔ (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
2019spcegv 3586 . . . . . . . . . . 11 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
21203ad2ant2 1131 . . . . . . . . . 10 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
2221adantr 479 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
23 vex 3477 . . . . . . . . . 10 𝑧 ∈ V
24 brcog 5873 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑋𝐴) → (𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋 ↔ ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
2523, 12, 24sylancr 585 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → (𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋 ↔ ∃𝑦(𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
2622, 25sylibrd 258 . . . . . . . 8 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → 𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋))
27 simpl1 1188 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)
2827ssbrd 5195 . . . . . . . 8 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → (𝑧((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴)))𝑋𝑧𝑅𝑋))
2926, 28syld 47 . . . . . . 7 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋) → 𝑧𝑅𝑋))
3016, 29mpan2d 692 . . . . . 6 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑧𝑅𝑋))
317, 30biimtrrid 242 . . . . 5 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑧𝑅𝑌𝑧(𝐴 × 𝐴)𝑌) → 𝑧𝑅𝑋))
326, 31mpan2d 692 . . . 4 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → (𝑧𝑅𝑌𝑧𝑅𝑋))
3332imdistanda 570 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → ((𝑧𝐴𝑧𝑅𝑌) → (𝑧𝐴𝑧𝑅𝑋)))
3423elpred 6327 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧𝐴𝑧𝑅𝑌)))
35343ad2ant2 1131 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧𝐴𝑧𝑅𝑌)))
3623elpred 6327 . . . 4 (𝑋𝐴 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧𝐴𝑧𝑅𝑋)))
37363ad2ant3 1132 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧𝐴𝑧𝑅𝑋)))
3833, 35, 373imtr4d 293 . 2 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) → 𝑧 ∈ Pred(𝑅, 𝐴, 𝑋)))
3938ssrdv 3988 1 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  cin 3948  wss 3949   class class class wbr 5152   × cxp 5680  ccom 5686  Predcpred 6309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310
This theorem is referenced by:  predpo  6334  frmin  9782  frrlem16  9791
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