ressprs - Mathbox for Thierry Arnoux

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Theorem ressprs 32120

Description: The restriction of a proset is a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)

**Hypothesis**
Ref	Expression
ressprs.b	⊢ 𝐵 = (Base‘𝐾)

**Assertion**
Ref	Expression
ressprs	⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾_s 𝐴) ∈ Proset )

Proof of Theorem ressprs Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7440	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾_s 𝐴) ∈ V)
2		simp-4l 781	. . . . . . . 8 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝐾 ∈ Proset )
3		simp-4r 782	. . . . . . . . 9 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
4		simpllr 774	. . . . . . . . 9 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ 𝐴)
5	3, 4	sseldd 3982	. . . . . . . 8 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ 𝐵)
6	2, 5	jca 512	. . . . . . 7 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵))
7		simplr 767	. . . . . . . 8 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝐴)
8	3, 7	sseldd 3982	. . . . . . 7 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝐵)
9		simpr 485	. . . . . . . 8 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
10	3, 9	sseldd 3982	. . . . . . 7 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐵)
11		ressprs.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐾)
12		eqid 2732	. . . . . . . . . . . 12 ⊢ (le‘𝐾) = (le‘𝐾)
13	11, 12	isprs 18246	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
14	13	simprbi 497	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
15	14	r19.21bi 3248	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
16	15	r19.21bi 3248	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
17	16	r19.21bi 3248	. . . . . . 7 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
18	6, 8, 10, 17	syl21anc 836	. . . . . 6 ⊢ (((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
19	18	ralrimiva 3146	. . . . 5 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
20	19	ralrimiva 3146	. . . 4 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
21	20	ralrimiva 3146	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
22		eqid 2732	. . . . . . 7 ⊢ (𝐾 ↾_s 𝐴) = (𝐾 ↾_s 𝐴)
23	22, 11	ressbas2 17178	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾_s 𝐴)))
24	23	adantl 482	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘(𝐾 ↾_s 𝐴)))
25	11	fvexi 6902	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
26	25	ssex 5320	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
27	22, 12	ressle 17321	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾 ↾_s 𝐴)))
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → (le‘𝐾) = (le‘(𝐾 ↾_s 𝐴)))
29	28	adantl 482	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (le‘𝐾) = (le‘(𝐾 ↾_s 𝐴)))
30	29	breqd 5158	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥(le‘𝐾)𝑥 ↔ 𝑥(le‘(𝐾 ↾_s 𝐴))𝑥))
31	29	breqd 5158	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥(le‘𝐾)𝑦 ↔ 𝑥(le‘(𝐾 ↾_s 𝐴))𝑦))
32	29	breqd 5158	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑦(le‘𝐾)𝑧 ↔ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧))
33	31, 32	anbi12d 631	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) ↔ (𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧)))
34	29	breqd 5158	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥(le‘𝐾)𝑧 ↔ 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧))
35	33, 34	imbi12d 344	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧) ↔ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧)))
36	30, 35	anbi12d 631	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)) ↔ (𝑥(le‘(𝐾 ↾_s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧))))
37	24, 36	raleqbidv 3342	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (∀𝑧 ∈ 𝐴 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)) ↔ ∀𝑧 ∈ (Base‘(𝐾 ↾_s 𝐴))(𝑥(le‘(𝐾 ↾_s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧))))
38	24, 37	raleqbidv 3342	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)) ↔ ∀𝑦 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑧 ∈ (Base‘(𝐾 ↾_s 𝐴))(𝑥(le‘(𝐾 ↾_s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧))))
39	24, 38	raleqbidv 3342	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)) ↔ ∀𝑥 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑦 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑧 ∈ (Base‘(𝐾 ↾_s 𝐴))(𝑥(le‘(𝐾 ↾_s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧))))
40	39	anbi2d 629	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (((𝐾 ↾_s 𝐴) ∈ V ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))) ↔ ((𝐾 ↾_s 𝐴) ∈ V ∧ ∀𝑥 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑦 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑧 ∈ (Base‘(𝐾 ↾_s 𝐴))(𝑥(le‘(𝐾 ↾_s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧)))))
41	1, 21, 40	mpbi2and 710	. 2 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((𝐾 ↾_s 𝐴) ∈ V ∧ ∀𝑥 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑦 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑧 ∈ (Base‘(𝐾 ↾_s 𝐴))(𝑥(le‘(𝐾 ↾_s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧))))
42		eqid 2732	. . 3 ⊢ (Base‘(𝐾 ↾_s 𝐴)) = (Base‘(𝐾 ↾_s 𝐴))
43		eqid 2732	. . 3 ⊢ (le‘(𝐾 ↾_s 𝐴)) = (le‘(𝐾 ↾_s 𝐴))
44	42, 43	isprs 18246	. 2 ⊢ ((𝐾 ↾_s 𝐴) ∈ Proset ↔ ((𝐾 ↾_s 𝐴) ∈ V ∧ ∀𝑥 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑦 ∈ (Base‘(𝐾 ↾_s 𝐴))∀𝑧 ∈ (Base‘(𝐾 ↾_s 𝐴))(𝑥(le‘(𝐾 ↾_s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐾 ↾_s 𝐴))𝑦 ∧ 𝑦(le‘(𝐾 ↾_s 𝐴))𝑧) → 𝑥(le‘(𝐾 ↾_s 𝐴))𝑧))))
45	41, 44	sylibr 233	1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾_s 𝐴) ∈ Proset )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3947 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 ↾_s cress 17169 lecple 17200 Proset cproset 18242

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-dec 12674 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-ple 17213 df-proset 18244

This theorem is referenced by: prsssdm 32885 ordtrestNEW 32889 ordtrest2NEW 32891

W3C validator