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Theorem preddowncl 6355
Description: A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)
Assertion
Ref Expression
preddowncl ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem preddowncl
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2827 . . . . 5 (𝑦 = 𝑋 → (𝑦𝐵𝑋𝐵))
2 predeq3 6327 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑋))
3 predeq3 6327 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
42, 3eqeq12d 2751 . . . . 5 (𝑦 = 𝑋 → (Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦) ↔ Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
51, 4imbi12d 344 . . . 4 (𝑦 = 𝑋 → ((𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)) ↔ (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
65imbi2d 340 . . 3 (𝑦 = 𝑋 → (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))) ↔ ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))))
7 predpredss 6330 . . . . . 6 (𝐵𝐴 → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
87ad2antrr 726 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
9 predeq3 6327 . . . . . . . . . . . 12 (𝑥 = 𝑦 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑦))
109sseq1d 4027 . . . . . . . . . . 11 (𝑥 = 𝑦 → (Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
1110rspccva 3621 . . . . . . . . . 10 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
1211sseld 3994 . . . . . . . . 9 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝐵))
13 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
1413elpredim 6339 . . . . . . . . 9 (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦)
1512, 14jca2 513 . . . . . . . 8 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦)))
16 vex 3482 . . . . . . . . . . 11 𝑧 ∈ V
1716elpred 6340 . . . . . . . . . 10 (𝑦𝐵 → (𝑧 ∈ Pred(𝑅, 𝐵, 𝑦) ↔ (𝑧𝐵𝑧𝑅𝑦)))
1817imbi2d 340 . . . . . . . . 9 (𝑦𝐵 → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦))))
1918adantl 481 . . . . . . . 8 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦))))
2015, 19mpbird 257 . . . . . . 7 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)))
2120ssrdv 4001 . . . . . 6 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
2221adantll 714 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
238, 22eqssd 4013 . . . 4 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))
2423ex 412 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)))
256, 24vtoclg 3554 . 2 (𝑋𝐵 → ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
2625pm2.43b 55 1 ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963   class class class wbr 5148  Predcpred 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323
This theorem is referenced by:  frrlem4  8313  wfrlem4OLD  8351
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