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Theorem predtrss 6219
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 485 . . . . . 6 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑧𝐴)
2 predel 6217 . . . . . . . 8 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
323ad2ant2 1133 . . . . . . 7 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 𝑌𝐴)
43adantr 481 . . . . . 6 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌𝐴)
5 brxp 5632 . . . . . 6 (𝑧(𝐴 × 𝐴)𝑌 ↔ (𝑧𝐴𝑌𝐴))
61, 4, 5sylanbrc 583 . . . . 5 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑧(𝐴 × 𝐴)𝑌)
7 brin 5126 . . . . . 6 (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ↔ (𝑧𝑅𝑌𝑧(𝐴 × 𝐴)𝑌))
8 predbrg 6218 . . . . . . . . . . 11 ((𝑋𝐴𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
98ancoms 459 . . . . . . . . . 10 ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 𝑌𝑅𝑋)
1093adant1 1129 . . . . . . . . 9 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 𝑌𝑅𝑋)
1110adantr 481 . . . . . . . 8 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌𝑅𝑋)
12 simpl3 1192 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑋𝐴)
13 brxp 5632 . . . . . . . . 9 (𝑌(𝐴 × 𝐴)𝑋 ↔ (𝑌𝐴𝑋𝐴))
144, 12, 13sylanbrc 583 . . . . . . . 8 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌(𝐴 × 𝐴)𝑋)
15 brin 5126 . . . . . . . 8 (𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋 ↔ (𝑌𝑅𝑋𝑌(𝐴 × 𝐴)𝑋))
1611, 14, 15sylanbrc 583 . . . . . . 7 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋)
17 breq2 5078 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌))
18 breq1 5077 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋))
1917, 18anbi12d 631 . . . . . . . . . . . 12 (𝑦 = 𝑌 → ((𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑋) ↔ (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑌(𝑅 ∩ (𝐴 × 𝐴))𝑋)))
2019spcegv 3534 . . . . . . . . . . 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 1133 . . . . . . . . . 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 481 . . . . . . . . 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 3434 . . . . . . . . . 10 𝑧 ∈ V
24 brcog 5769 . . . . . . . . . 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 587 . . . . . . . . 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 1190 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)
2827ssbrd 5117 . . . . . . . 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 691 . . . . . 6 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌𝑧𝑅𝑋))
317, 30syl5bir 242 . . . . 5 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑧𝑅𝑌𝑧(𝐴 × 𝐴)𝑌) → 𝑧𝑅𝑋))
326, 31mpan2d 691 . . . 4 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → (𝑧𝑅𝑌𝑧𝑅𝑋))
3332imdistanda 572 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → ((𝑧𝐴𝑧𝑅𝑌) → (𝑧𝐴𝑧𝑅𝑋)))
3423elpred 6213 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧𝐴𝑧𝑅𝑌)))
35343ad2ant2 1133 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧𝐴𝑧𝑅𝑌)))
3623elpred 6213 . . . 4 (𝑋𝐴 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧𝐴𝑧𝑅𝑋)))
37363ad2ant3 1134 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧𝐴𝑧𝑅𝑋)))
3833, 35, 373imtr4d 294 . 2 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) → 𝑧 ∈ Pred(𝑅, 𝐴, 𝑋)))
3938ssrdv 3927 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 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  Vcvv 3430  cin 3886  wss 3887   class class class wbr 5074   × cxp 5583  ccom 5589  Predcpred 6195
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5075  df-opab 5137  df-xp 5591  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196
This theorem is referenced by:  predpo  6220  frmin  9495  frrlem16  9504
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