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Theorem preddowncl 5927
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 2880 . . . . 5 (𝑦 = 𝑋 → (𝑦𝐵𝑋𝐵))
2 predeq3 5904 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑋))
3 predeq3 5904 . . . . . 6 (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
42, 3eqeq12d 2828 . . . . 5 (𝑦 = 𝑋 → (Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦) ↔ Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
51, 4imbi12d 335 . . . 4 (𝑦 = 𝑋 → ((𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)) ↔ (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
65imbi2d 331 . . 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 5906 . . . . . 6 (𝐵𝐴 → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
87ad2antrr 708 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
9 predeq3 5904 . . . . . . . . . . . 12 (𝑥 = 𝑦 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑦))
109sseq1d 3836 . . . . . . . . . . 11 (𝑥 = 𝑦 → (Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
1110rspccva 3508 . . . . . . . . . 10 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
1211sseld 3804 . . . . . . . . 9 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝐵))
13 vex 3401 . . . . . . . . . . 11 𝑦 ∈ V
1413elpredim 5912 . . . . . . . . . 10 (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦)
1514a1i 11 . . . . . . . . 9 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦))
1612, 15jcad 504 . . . . . . . 8 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦)))
17 vex 3401 . . . . . . . . . . 11 𝑧 ∈ V
1817elpred 5913 . . . . . . . . . 10 (𝑦𝐵 → (𝑧 ∈ Pred(𝑅, 𝐵, 𝑦) ↔ (𝑧𝐵𝑧𝑅𝑦)))
1918imbi2d 331 . . . . . . . . 9 (𝑦𝐵 → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦))))
2019adantl 469 . . . . . . . 8 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧𝐵𝑧𝑅𝑦))))
2116, 20mpbird 248 . . . . . . 7 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)))
2221ssrdv 3811 . . . . . 6 ((∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
2322adantll 696 . . . . 5 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
248, 23eqssd 3822 . . . 4 (((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))
2524ex 399 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)))
266, 25vtoclg 3466 . 2 (𝑋𝐵 → ((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
2726pm2.43b 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 197  wa 384   = wceq 1637  wcel 2157  wral 3103  wss 3776   class class class wbr 4851  Predcpred 5899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-xp 5324  df-cnv 5326  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900
This theorem is referenced by:  wfrlem4  7656  wfrlem4OLD  7657  frrlem4  32109
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