Theorem rrxsnicc 40520

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 7914	. . . . . 6 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 Fn 𝑋)
2	1	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 Fn 𝑋)
3		rrxsnicc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑𝑚 𝑋))
4		elmapfn 7880	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑𝑚 𝑋) → 𝐴 Fn 𝑋)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 Fn 𝑋)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝐴 Fn 𝑋)
7		simpll 790	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝜑)
8		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
9	8, 8	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = ((𝐴‘𝑗)[,](𝐴‘𝑗)))
10	9	cbvixpv 7926	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))
11	10	eleq2i 2693	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
12	11	biimpi 206	. . . . . . 7 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
13	12	ad2antlr 763	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
14		simpr 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
15		elmapi 7879	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (ℝ ↑𝑚 𝑋) → 𝐴:𝑋⟶ℝ)
16	3, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
18	17	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
19	18, 18	iccssred 39727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → ((𝐴‘𝑗)[,](𝐴‘𝑗)) ⊆ ℝ)
20		fvixp2 39389	. . . . . . . . . 10 ⊢ ((𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
21	20	adantll 750	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
22	19, 21	sseldd 3604	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ℝ)
23	22	rexrd 10089	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ℝ*)
24	18	rexrd 10089	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ*)
25		iccleub 12229	. . . . . . . 8 ⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝑓‘𝑗) ≤ (𝐴‘𝑗))
26	24, 24, 21, 25	syl3anc 1326	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ≤ (𝐴‘𝑗))
27		iccgelb 12230	. . . . . . . 8 ⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝐴‘𝑗) ≤ (𝑓‘𝑗))
28	24, 24, 21, 27	syl3anc 1326	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ≤ (𝑓‘𝑗))
29	23, 24, 26, 28	xrletrid 11986	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗))
30	7, 13, 14, 29	syl21anc 1325	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗))
31	2, 6, 30	eqfnfvd 6314	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 = 𝐴)
32		velsn 4193	. . . . . 6 ⊢ (𝑓 ∈ {𝐴} ↔ 𝑓 = 𝐴)
33	32	bicomi 214	. . . . 5 ⊢ (𝑓 = 𝐴 ↔ 𝑓 ∈ {𝐴})
34	33	biimpi 206	. . . 4 ⊢ (𝑓 = 𝐴 → 𝑓 ∈ {𝐴})
35	31, 34	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 ∈ {𝐴})
36	35	ssd 39252	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ⊆ {𝐴})
37	3	elexd 3214	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
38	16	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
39	38	leidd 10594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
40	38, 38, 38, 39, 39	eliccd 39726	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
41	40	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
42	37, 5, 41	3jca 1242	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘))))
43		elixp2 7912	. . . 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 224	. . 3 ⊢ (𝜑 → 𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
45		snssg 4327	. . . 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 222	. 2 ⊢ (𝜑 → {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
48	36, 47	eqssd 3620	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  {csn 4177   class class class wbr 4653   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Xcixp 7908  ℝcr 9935  ℝ^*cxr 10073   ≤ cle 10075  [,]cicc 12178

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182

This theorem is referenced by:  snvonmbl  40900  vonsn  40905