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Theorem preddowncl 6143
 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 2877 . . . . 5 (𝑦 = 𝑋 → (𝑦𝐵𝑋𝐵))
2 predeq3 6120 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑋))
3 predeq3 6120 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
42, 3eqeq12d 2814 . . . . 5 (𝑦 = 𝑋 → (Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦) ↔ Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
51, 4imbi12d 348 . . . 4 (𝑦 = 𝑋 → ((𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)) ↔ (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
65imbi2d 344 . . 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 6122 . . . . . 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 6120 . . . . . . . . . . . 12 (𝑥 = 𝑦 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑦))
109sseq1d 3946 . . . . . . . . . . 11 (𝑥 = 𝑦 → (Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
1110rspccva 3570 . . . . . . . . . 10 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
1211sseld 3914 . . . . . . . . 9 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝐵))
13 vex 3444 . . . . . . . . . 10 𝑦 ∈ V
1413elpredim 6128 . . . . . . . . 9 (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦)
1512, 14jca2 517 . . . . . . . 8 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦)))
16 vex 3444 . . . . . . . . . . 11 𝑧 ∈ V
1716elpred 6129 . . . . . . . . . 10 (𝑦𝐵 → (𝑧 ∈ Pred(𝑅, 𝐵, 𝑦) ↔ (𝑧𝐵𝑧𝑅𝑦)))
1817imbi2d 344 . . . . . . . . 9 (𝑦𝐵 → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦))))
1918adantl 485 . . . . . . . 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 260 . . . . . . 7 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)))
2120ssrdv 3921 . . . . . 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 3932 . . . 4 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))
2423ex 416 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)))
256, 24vtoclg 3515 . 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881   class class class wbr 5030  Predcpred 6115 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116 This theorem is referenced by:  wfrlem4  7943  frrlem4  33251
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