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Theorem rrxsnicc 40289
Description: A multidimensional singleton expressed as a multidimensional closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
rrxsnicc.1 (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))
Assertion
Ref Expression
rrxsnicc (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = {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.
StepHypRef Expression
1 ixpfn 7911 . . . . . 6 (𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) → 𝑓 Fn 𝑋)
21adantl 482 . . . . 5 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝑓 Fn 𝑋)
3 rrxsnicc.1 . . . . . . 7 (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))
4 elmapfn 7877 . . . . . . 7 (𝐴 ∈ (ℝ ↑𝑚 𝑋) → 𝐴 Fn 𝑋)
53, 4syl 17 . . . . . 6 (𝜑𝐴 Fn 𝑋)
65adantr 481 . . . . 5 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝐴 Fn 𝑋)
7 simpll 790 . . . . . 6 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → 𝜑)
8 fveq2 6189 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
98, 8oveq12d 6665 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝐴𝑘)[,](𝐴𝑘)) = ((𝐴𝑗)[,](𝐴𝑗)))
109cbvixpv 7923 . . . . . . . . 9 X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))
1110eleq2i 2692 . . . . . . . 8 (𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ↔ 𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)))
1211biimpi 206 . . . . . . 7 (𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) → 𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)))
1312ad2antlr 763 . . . . . 6 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → 𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)))
14 simpr 477 . . . . . 6 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → 𝑗𝑋)
15 elmapi 7876 . . . . . . . . . . . . 13 (𝐴 ∈ (ℝ ↑𝑚 𝑋) → 𝐴:𝑋⟶ℝ)
163, 15syl 17 . . . . . . . . . . . 12 (𝜑𝐴:𝑋⟶ℝ)
1716ffvelrnda 6357 . . . . . . . . . . 11 ((𝜑𝑗𝑋) → (𝐴𝑗) ∈ ℝ)
1817adantlr 751 . . . . . . . . . 10 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝐴𝑗) ∈ ℝ)
1918, 18iccssred 39536 . . . . . . . . 9 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → ((𝐴𝑗)[,](𝐴𝑗)) ⊆ ℝ)
20 fvixp2 39211 . . . . . . . . . 10 ((𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗)))
2120adantll 750 . . . . . . . . 9 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗)))
2219, 21sseldd 3602 . . . . . . . 8 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ℝ)
2322rexrd 10086 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ℝ*)
2418rexrd 10086 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝐴𝑗) ∈ ℝ*)
25 iccleub 12226 . . . . . . . 8 (((𝐴𝑗) ∈ ℝ* ∧ (𝐴𝑗) ∈ ℝ* ∧ (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗))) → (𝑓𝑗) ≤ (𝐴𝑗))
2624, 24, 21, 25syl3anc 1325 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ≤ (𝐴𝑗))
27 iccgelb 12227 . . . . . . . 8 (((𝐴𝑗) ∈ ℝ* ∧ (𝐴𝑗) ∈ ℝ* ∧ (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗))) → (𝐴𝑗) ≤ (𝑓𝑗))
2824, 24, 21, 27syl3anc 1325 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝐴𝑗) ≤ (𝑓𝑗))
2923, 24, 26, 28xrletrid 11983 . . . . . 6 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) = (𝐴𝑗))
307, 13, 14, 29syl21anc 1324 . . . . 5 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → (𝑓𝑗) = (𝐴𝑗))
312, 6, 30eqfnfvd 6312 . . . 4 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝑓 = 𝐴)
32 velsn 4191 . . . . . 6 (𝑓 ∈ {𝐴} ↔ 𝑓 = 𝐴)
3332bicomi 214 . . . . 5 (𝑓 = 𝐴𝑓 ∈ {𝐴})
3433biimpi 206 . . . 4 (𝑓 = 𝐴𝑓 ∈ {𝐴})
3531, 34syl 17 . . 3 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝑓 ∈ {𝐴})
3635ssd 39078 . 2 (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ⊆ {𝐴})
373elexd 3212 . . . . 5 (𝜑𝐴 ∈ V)
3816ffvelrnda 6357 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3938leidd 10591 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐴𝑘))
4038, 38, 38, 39, 39eliccd 39535 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘)))
4140ralrimiva 2965 . . . . 5 (𝜑 → ∀𝑘𝑋 (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘)))
4237, 5, 413jca 1241 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘𝑋 (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘))))
43 elixp2 7909 . . . 4 (𝐴X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ↔ (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘𝑋 (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘))))
4442, 43sylibr 224 . . 3 (𝜑𝐴X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)))
45 snssg 4325 . . . 4 (𝐴 ∈ (ℝ ↑𝑚 𝑋) → (𝐴X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ↔ {𝐴} ⊆ X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))))
463, 45syl 17 . . 3 (𝜑 → (𝐴X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ↔ {𝐴} ⊆ X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))))
4744, 46mpbid 222 . 2 (𝜑 → {𝐴} ⊆ X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)))
4836, 47eqssd 3618 1 (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  Vcvv 3198  wss 3572  {csn 4175   class class class wbr 4651   Fn wfn 5881  wf 5882  cfv 5886  (class class class)co 6647  𝑚 cmap 7854  Xcixp 7905  cr 9932  *cxr 10070  cle 10072  [,]cicc 12175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-pre-lttri 10007  ax-pre-lttrn 10008
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-po 5033  df-so 5034  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-er 7739  df-map 7856  df-ixp 7906  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-icc 12179
This theorem is referenced by:  snvonmbl  40669  vonsn  40674
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