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Theorem rrxsnicc 42022
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 8265 . . . . . 6 (𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) → 𝑓 Fn 𝑋)
21adantl 474 . . . . 5 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝑓 Fn 𝑋)
3 rrxsnicc.1 . . . . . . 7 (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))
4 elmapfn 8229 . . . . . . 7 (𝐴 ∈ (ℝ ↑𝑚 𝑋) → 𝐴 Fn 𝑋)
53, 4syl 17 . . . . . 6 (𝜑𝐴 Fn 𝑋)
65adantr 473 . . . . 5 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝐴 Fn 𝑋)
7 simpll 754 . . . . . 6 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → 𝜑)
8 fveq2 6499 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
98, 8oveq12d 6994 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝐴𝑘)[,](𝐴𝑘)) = ((𝐴𝑗)[,](𝐴𝑗)))
109cbvixpv 8277 . . . . . . . . 9 X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))
1110eleq2i 2857 . . . . . . . 8 (𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ↔ 𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)))
1211biimpi 208 . . . . . . 7 (𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) → 𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)))
1312ad2antlr 714 . . . . . 6 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → 𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)))
14 simpr 477 . . . . . 6 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → 𝑗𝑋)
15 elmapi 8228 . . . . . . . . . . . . 13 (𝐴 ∈ (ℝ ↑𝑚 𝑋) → 𝐴:𝑋⟶ℝ)
163, 15syl 17 . . . . . . . . . . . 12 (𝜑𝐴:𝑋⟶ℝ)
1716ffvelrnda 6676 . . . . . . . . . . 11 ((𝜑𝑗𝑋) → (𝐴𝑗) ∈ ℝ)
1817adantlr 702 . . . . . . . . . 10 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝐴𝑗) ∈ ℝ)
1918, 18iccssred 41217 . . . . . . . . 9 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → ((𝐴𝑗)[,](𝐴𝑗)) ⊆ ℝ)
20 fvixp2 40892 . . . . . . . . . 10 ((𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗)) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗)))
2120adantll 701 . . . . . . . . 9 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗)))
2219, 21sseldd 3859 . . . . . . . 8 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ℝ)
2322rexrd 10490 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ∈ ℝ*)
2418rexrd 10490 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝐴𝑗) ∈ ℝ*)
25 iccleub 12608 . . . . . . . 8 (((𝐴𝑗) ∈ ℝ* ∧ (𝐴𝑗) ∈ ℝ* ∧ (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗))) → (𝑓𝑗) ≤ (𝐴𝑗))
2624, 24, 21, 25syl3anc 1351 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) ≤ (𝐴𝑗))
27 iccgelb 12609 . . . . . . . 8 (((𝐴𝑗) ∈ ℝ* ∧ (𝐴𝑗) ∈ ℝ* ∧ (𝑓𝑗) ∈ ((𝐴𝑗)[,](𝐴𝑗))) → (𝐴𝑗) ≤ (𝑓𝑗))
2824, 24, 21, 27syl3anc 1351 . . . . . . 7 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝐴𝑗) ≤ (𝑓𝑗))
2923, 24, 26, 28xrletrid 12365 . . . . . 6 (((𝜑𝑓X𝑗𝑋 ((𝐴𝑗)[,](𝐴𝑗))) ∧ 𝑗𝑋) → (𝑓𝑗) = (𝐴𝑗))
307, 13, 14, 29syl21anc 825 . . . . 5 (((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) ∧ 𝑗𝑋) → (𝑓𝑗) = (𝐴𝑗))
312, 6, 30eqfnfvd 6630 . . . 4 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝑓 = 𝐴)
32 velsn 4457 . . . . . 6 (𝑓 ∈ {𝐴} ↔ 𝑓 = 𝐴)
3332bicomi 216 . . . . 5 (𝑓 = 𝐴𝑓 ∈ {𝐴})
3433biimpi 208 . . . 4 (𝑓 = 𝐴𝑓 ∈ {𝐴})
3531, 34syl 17 . . 3 ((𝜑𝑓X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘))) → 𝑓 ∈ {𝐴})
3635ssd 40769 . 2 (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ⊆ {𝐴})
373elexd 3435 . . . . 5 (𝜑𝐴 ∈ V)
3816ffvelrnda 6676 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3938leidd 11007 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐴𝑘))
4038, 38, 38, 39, 39eliccd 41216 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘)))
4140ralrimiva 3132 . . . . 5 (𝜑 → ∀𝑘𝑋 (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘)))
4237, 5, 413jca 1108 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘𝑋 (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘))))
43 elixp2 8263 . . . 4 (𝐴X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) ↔ (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘𝑋 (𝐴𝑘) ∈ ((𝐴𝑘)[,](𝐴𝑘))))
4442, 43sylibr 226 . . 3 (𝜑𝐴X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)))
45 snssg 4591 . . . 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 224 . 2 (𝜑 → {𝐴} ⊆ X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)))
4836, 47eqssd 3875 1 (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3088  Vcvv 3415  wss 3829  {csn 4441   class class class wbr 4929   Fn wfn 6183  wf 6184  cfv 6188  (class class class)co 6976  𝑚 cmap 8206  Xcixp 8259  cr 10334  *cxr 10473  cle 10475  [,]cicc 12557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-pre-lttri 10409  ax-pre-lttrn 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-po 5326  df-so 5327  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-er 8089  df-map 8208  df-ixp 8260  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-icc 12561
This theorem is referenced by:  snvonmbl  42405  vonsn  42410
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