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Theorem preddowncl 6287
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 2822 . . . . 5 (𝑦 = 𝑋 → (𝑦𝐵𝑋𝐵))
2 predeq3 6258 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑋))
3 predeq3 6258 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
42, 3eqeq12d 2749 . . . . 5 (𝑦 = 𝑋 → (Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦) ↔ Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
51, 4imbi12d 345 . . . 4 (𝑦 = 𝑋 → ((𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)) ↔ (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
65imbi2d 341 . . 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 6261 . . . . . 6 (𝐵𝐴 → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
87ad2antrr 725 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
9 predeq3 6258 . . . . . . . . . . . 12 (𝑥 = 𝑦 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑦))
109sseq1d 3976 . . . . . . . . . . 11 (𝑥 = 𝑦 → (Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
1110rspccva 3579 . . . . . . . . . 10 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
1211sseld 3944 . . . . . . . . 9 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝐵))
13 vex 3448 . . . . . . . . . 10 𝑦 ∈ V
1413elpredim 6270 . . . . . . . . 9 (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦)
1512, 14jca2 515 . . . . . . . 8 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦)))
16 vex 3448 . . . . . . . . . . 11 𝑧 ∈ V
1716elpred 6271 . . . . . . . . . 10 (𝑦𝐵 → (𝑧 ∈ Pred(𝑅, 𝐵, 𝑦) ↔ (𝑧𝐵𝑧𝑅𝑦)))
1817imbi2d 341 . . . . . . . . 9 (𝑦𝐵 → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦))))
1918adantl 483 . . . . . . . 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 3951 . . . . . 6 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
2221adantll 713 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
238, 22eqssd 3962 . . . 4 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))
2423ex 414 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)))
256, 24vtoclg 3524 . 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 205  wa 397   = wceq 1542  wcel 2107  wral 3061  wss 3911   class class class wbr 5106  Predcpred 6253
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254
This theorem is referenced by:  frrlem4  8221  wfrlem4OLD  8259
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