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Theorem predtrss 6276
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 6274 . . . . . . . 8 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
323ad2ant2 1134 . . . . . . 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 5681 . . . . . 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 5157 . . . . . 6 (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌 ↔ (𝑧𝑅𝑌𝑧(𝐴 × 𝐴)𝑌))
8 predbrg 6275 . . . . . . . . . . 11 ((𝑋𝐴𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
98ancoms 459 . . . . . . . . . 10 ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → 𝑌𝑅𝑋)
1093adant1 1130 . . . . . . . . 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 1193 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → 𝑋𝐴)
13 brxp 5681 . . . . . . . . 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 5157 . . . . . . . 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 5109 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑌))
18 breq1 5108 . . . . . . . . . . . . 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 3556 . . . . . . . . . . 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 1134 . . . . . . . . . 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 3449 . . . . . . . . . 10 𝑧 ∈ V
24 brcog 5822 . . . . . . . . . 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 1191 . . . . . . . . 9 (((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) ∧ 𝑧𝐴) → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)
2827ssbrd 5148 . . . . . . . 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 572 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → ((𝑧𝐴𝑧𝑅𝑌) → (𝑧𝐴𝑧𝑅𝑋)))
3423elpred 6270 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧𝐴𝑧𝑅𝑌)))
35343ad2ant2 1134 . . 3 ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑌) ↔ (𝑧𝐴𝑧𝑅𝑌)))
3623elpred 6270 . . . 4 (𝑋𝐴 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑧𝐴𝑧𝑅𝑋)))
37363ad2ant3 1135 . . 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 3950 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 1087   = wceq 1541  wex 1781  wcel 2106  Vcvv 3445  cin 3909  wss 3910   class class class wbr 5105   × cxp 5631  ccom 5637  Predcpred 6252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253
This theorem is referenced by:  predpo  6277  frmin  9685  frrlem16  9694
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