Theorem rrxsnicc 46757

Description: A multidimensional singleton expressed as a multidimensional closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypothesis

Ref	Expression
rrxsnicc.1	⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))

Assertion

Ref	Expression
rrxsnicc	⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})

Distinct variable groups:   𝐴,𝑘   𝑘,𝑋   𝜑,𝑘

Proof of Theorem rrxsnicc

Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpfn 8845	. . . . . 6 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 Fn 𝑋)
2	1	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 Fn 𝑋)
3		rrxsnicc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))
4		elmapfn 8806	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → 𝐴 Fn 𝑋)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 Fn 𝑋)
6	5	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝐴 Fn 𝑋)
7		simpll 773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝜑)
8		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
9	8, 8	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = ((𝐴‘𝑗)[,](𝐴‘𝑗)))
10	9	cbvixpv 8857	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))
11	10	eleq2i 2833	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
12	11	biimpi 218	. . . . . . 7 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
13	12	ad2antlr 734	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
14		simpr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
15		elmapi 8790	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → 𝐴:𝑋⟶ℝ)
16	3, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	ffvelcdmda 7029	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
18	17	adantlr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
19	18, 18	iccssred 13382	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → ((𝐴‘𝑗)[,](𝐴‘𝑗)) ⊆ ℝ)
20		fvixp2 45659	. . . . . . . . . 10 ⊢ ((𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
21	20	adantll 721	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
22	19, 21	sseldd 3918	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ℝ)
23	22	rexrd 11190	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ℝ*)
24	18	rexrd 11190	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ*)
25		iccleub 13349	. . . . . . . 8 ⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝑓‘𝑗) ≤ (𝐴‘𝑗))
26	24, 24, 21, 25	syl3anc 1380	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ≤ (𝐴‘𝑗))
27		iccgelb 13350	. . . . . . . 8 ⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝐴‘𝑗) ≤ (𝑓‘𝑗))
28	24, 24, 21, 27	syl3anc 1380	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ≤ (𝑓‘𝑗))
29	23, 24, 26, 28	xrletrid 13101	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗))
30	7, 13, 14, 29	syl21anc 844	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗))
31	2, 6, 30	eqfnfvd 6978	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 = 𝐴)
32		velsn 4574	. . . . . 6 ⊢ (𝑓 ∈ {𝐴} ↔ 𝑓 = 𝐴)
33	32	bicomi 226	. . . . 5 ⊢ (𝑓 = 𝐴 ↔ 𝑓 ∈ {𝐴})
34	33	biimpi 218	. . . 4 ⊢ (𝑓 = 𝐴 → 𝑓 ∈ {𝐴})
35	31, 34	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 ∈ {𝐴})
36	35	ssd 45543	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ⊆ {𝐴})
37	3	elexd 3456	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
38	16	ffvelcdmda 7029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
39	38	leidd 11711	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
40	38, 38, 38, 39, 39	eliccd 45963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
41	40	ralrimiva 3133	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
42	37, 5, 41	3jca 1135	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘))))
43		elixp2 8843	. . . 4 ⊢ (𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘))))
44	42, 43	sylibr 236	. . 3 ⊢ (𝜑 → 𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
45		snssg 4718	. . . 4 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → (𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))))
46	3, 45	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))))
47	44, 46	mpbid 234	. 2 ⊢ (𝜑 → {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
48	36, 47	eqssd 3934	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ⊆ wss 3885  {csn 4558   class class class wbr 5075   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360   ↑_m cmap 8767  Xcixp 8839  ℝcr 11032  ℝ^*cxr 11173   ≤ cle 11175  [,]cicc 13296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-pre-lttri 11107  ax-pre-lttrn 11108

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-po 5529  df-so 5530  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-icc 13300

This theorem is referenced by:  snvonmbl  47143  vonsn  47148