Theorem rrxsnicc 46750

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 8846	. . . . . 6 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 Fn 𝑋)
2	1	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 Fn 𝑋)
3		rrxsnicc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))
4		elmapfn 8807	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → 𝐴 Fn 𝑋)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 Fn 𝑋)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝐴 Fn 𝑋)
7		simpll 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝜑)
8		fveq2 6836	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
9	8, 8	oveq12d 7380	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = ((𝐴‘𝑗)[,](𝐴‘𝑗)))
10	9	cbvixpv 8858	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))
11	10	eleq2i 2829	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
12	11	biimpi 216	. . . . . . 7 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
13	12	ad2antlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
14		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
15		elmapi 8791	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → 𝐴:𝑋⟶ℝ)
16	3, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	ffvelcdmda 7032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
18	17	adantlr 716	. . . . . . . . . 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 45650	. . . . . . . . . 10 ⊢ ((𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
21	20	adantll 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
22	19, 21	sseldd 3923	. . . . . . . 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 1374	. . . . . . 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 1374	. . . . . . 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 838	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗))
31	2, 6, 30	eqfnfvd 6982	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 = 𝐴)
32		velsn 4584	. . . . . 6 ⊢ (𝑓 ∈ {𝐴} ↔ 𝑓 = 𝐴)
33	32	bicomi 224	. . . . 5 ⊢ (𝑓 = 𝐴 ↔ 𝑓 ∈ {𝐴})
34	33	biimpi 216	. . . 4 ⊢ (𝑓 = 𝐴 → 𝑓 ∈ {𝐴})
35	31, 34	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 ∈ {𝐴})
36	35	ssd 45533	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ⊆ {𝐴})
37	3	elexd 3454	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
38	16	ffvelcdmda 7032	. . . . . . 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 45956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
41	40	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
42	37, 5, 41	3jca 1129	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘))))
43		elixp2 8844	. . . 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 234	. . 3 ⊢ (𝜑 → 𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
45		snssg 4728	. . . 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 232	. 2 ⊢ (𝜑 → {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
48	36, 47	eqssd 3940	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  {csn 4568   class class class wbr 5086   Fn wfn 6489  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ↑_m cmap 8768  Xcixp 8840  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-pre-lttri 11107  ax-pre-lttrn 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-po 5534  df-so 5535  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-er 8638  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-icc 13300

This theorem is referenced by:  snvonmbl  47136  vonsn  47141