Theorem preddowncl 5707

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 2689	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
2		predeq3 5684	. . . . . 6 ⊢ (𝑦 = 𝑋 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑋))
3		predeq3 5684	. . . . . 6 ⊢ (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
4	2, 3	eqeq12d 2637	. . . . 5 ⊢ (𝑦 = 𝑋 → (Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦) ↔ Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
5	1, 4	imbi12d 334	. . . 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 330	. . 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 5686	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
8	7	ad2antrr 762	. . . . 5 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑦))
9		predeq3 5684	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑦))
10	9	sseq1d 3632	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
11	10	rspccva 3308	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
12	11	sseld 3602	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ 𝐵))
13		vex 3203	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
14	13	elpredim 5692	. . . . . . . . . 10 ⊢ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦)
15	14	a1i 11	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧𝑅𝑦))
16	12, 15	jcad 555	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧 ∈ 𝐵 ∧ 𝑧𝑅𝑦)))
17		vex 3203	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
18	17	elpred 5693	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (𝑧 ∈ Pred(𝑅, 𝐵, 𝑦) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧𝑅𝑦)))
19	18	imbi2d 330	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ((𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)) ↔ (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → (𝑧 ∈ 𝐵 ∧ 𝑧𝑅𝑦))))
20	19	adantl 482	. . . . . . . 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))))
21	16, 20	mpbird 247	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑦) → 𝑧 ∈ Pred(𝑅, 𝐵, 𝑦)))
22	21	ssrdv 3609	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵 ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
23	22	adantll 750	. . . . 5 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐵, 𝑦))
24	8, 23	eqssd 3620	. . . 4 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) ∧ 𝑦 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦))
25	24	ex 450	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑦 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐴, 𝑦)))
26	6, 25	vtoclg 3266	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))))
27	26	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574   class class class wbr 4653  Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  wfrlem4  7418  frrlem4  31783