Theorem rrxsnicc 45593

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 8899	. . . . . 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 8861	. . . . . . 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 764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝜑)
8		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
9	8, 8	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = ((𝐴‘𝑗)[,](𝐴‘𝑗)))
10	9	cbvixpv 8911	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))
11	10	eleq2i 2819	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
12	11	biimpi 215	. . . . . . 7 ⊢ (𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)))
13	12	ad2antlr 724	. . . . . 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 8845	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → 𝐴:𝑋⟶ℝ)
16	3, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	ffvelcdmda 7080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
18	17	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
19	18, 18	iccssred 13417	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → ((𝐴‘𝑗)[,](𝐴‘𝑗)) ⊆ ℝ)
20		fvixp2 44475	. . . . . . . . . 10 ⊢ ((𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
21	20	adantll 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗)))
22	19, 21	sseldd 3978	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ℝ)
23	22	rexrd 11268	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ℝ*)
24	18	rexrd 11268	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ*)
25		iccleub 13385	. . . . . . . 8 ⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝑓‘𝑗) ≤ (𝐴‘𝑗))
26	24, 24, 21, 25	syl3anc 1368	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ≤ (𝐴‘𝑗))
27		iccgelb 13386	. . . . . . . 8 ⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝐴‘𝑗) ≤ (𝑓‘𝑗))
28	24, 24, 21, 27	syl3anc 1368	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ≤ (𝑓‘𝑗))
29	23, 24, 26, 28	xrletrid 13140	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗))
30	7, 13, 14, 29	syl21anc 835	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗))
31	2, 6, 30	eqfnfvd 7029	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 = 𝐴)
32		velsn 4639	. . . . . 6 ⊢ (𝑓 ∈ {𝐴} ↔ 𝑓 = 𝐴)
33	32	bicomi 223	. . . . 5 ⊢ (𝑓 = 𝐴 ↔ 𝑓 ∈ {𝐴})
34	33	biimpi 215	. . . 4 ⊢ (𝑓 = 𝐴 → 𝑓 ∈ {𝐴})
35	31, 34	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 ∈ {𝐴})
36	35	ssd 44349	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ⊆ {𝐴})
37	3	elexd 3489	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
38	16	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
39	38	leidd 11784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
40	38, 38, 38, 39, 39	eliccd 44794	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
41	40	ralrimiva 3140	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))
42	37, 5, 41	3jca 1125	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘))))
43		elixp2 8897	. . . 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 233	. . 3 ⊢ (𝜑 → 𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
45		snssg 4782	. . . 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 231	. 2 ⊢ (𝜑 → {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))
48	36, 47	eqssd 3994	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  Vcvv 3468   ⊆ wss 3943  {csn 4623   class class class wbr 5141   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7405   ↑_m cmap 8822  Xcixp 8893  ℝcr 11111  ℝ^*cxr 11251   ≤ cle 11253  [,]cicc 13333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-icc 13337

This theorem is referenced by:  snvonmbl  45979  vonsn  45984