Theorem preddowncl 6334

Description: A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)

Assertion

Ref	Expression
preddowncl	⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋 ∈ 𝐵 → 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.

Step	Hyp	Ref	Expression
1		eleq1 2822	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
2		predeq3 6305	. . . . . 6 ⊢ (𝑦 = 𝑋 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑋))
3		predeq3 6305	. . . . . 6 ⊢ (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
4	2, 3	eqeq12d 2749	. . . . 5 ⊢ (𝑦 = 𝑋 → (Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦) ↔ Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
5	1, 4	imbi12d 345	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)) ↔ (𝑋 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
6	5	imbi2d 341	. . 3 ⊢ (𝑦 = 𝑋 → (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))) ↔ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))))
7		predpredss 6308	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
8	7	ad2antrr 725	. . . . 5 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
9		predeq3 6305	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑦))
10	9	sseq1d 4014	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
11	10	rspccva 3612	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
12	11	sseld 3982	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ 𝐵))
13		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
14	13	elpredim 6317	. . . . . . . . 9 ⊢ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦)
15	12, 14	jca2 515	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧 ∈ 𝐵 ∧ 𝑧𝑅𝑦)))
16		vex 3479	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
17	16	elpred 6318	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (𝑧 ∈ Pred(𝑅, 𝐵, 𝑦) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧𝑅𝑦)))
18	17	imbi2d 341	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧 ∈ 𝐵 ∧ 𝑧𝑅𝑦))))
19	18	adantl 483	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧 ∈ 𝐵 ∧ 𝑧𝑅𝑦))))
20	15, 19	mpbird 257	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)))
21	20	ssrdv 3989	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
22	21	adantll 713	. . . . 5 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
23	8, 22	eqssd 4000	. . . 4 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))
24	23	ex 414	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)))
25	6, 24	vtoclg 3557	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
26	25	pm2.43b 55	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949   class class class wbr 5149  Predcpred 6300

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 5300  ax-nul 5307  ax-pr 5428

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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301

This theorem is referenced by:  frrlem4  8274  wfrlem4OLD  8312