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Theorem metustsym 22283
Description: Elements of the filter base generated by the metric 𝐷 are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
Assertion
Ref Expression
metustsym ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 = 𝐴)
Distinct variable groups:   𝐷,𝑎   𝑋,𝑎   𝐴,𝑎   𝐹,𝑎

Proof of Theorem metustsym
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metust.1 . . . 4 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
21metustss 22279 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
3 cnvss 5259 . . . 4 (𝐴 ⊆ (𝑋 × 𝑋) → 𝐴(𝑋 × 𝑋))
4 cnvxp 5515 . . . 4 (𝑋 × 𝑋) = (𝑋 × 𝑋)
53, 4syl6sseq 3635 . . 3 (𝐴 ⊆ (𝑋 × 𝑋) → 𝐴 ⊆ (𝑋 × 𝑋))
62, 5syl 17 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
7 simp-4l 805 . . . . . . . . . 10 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
8 simpr1r 1117 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ((𝑝𝑋𝑞𝑋) ∧ 𝑎 ∈ ℝ+𝐴 = (𝐷 “ (0[,)𝑎)))) → 𝑞𝑋)
983anassrs 1287 . . . . . . . . . 10 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝑞𝑋)
10 simpr1l 1116 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ((𝑝𝑋𝑞𝑋) ∧ 𝑎 ∈ ℝ+𝐴 = (𝐷 “ (0[,)𝑎)))) → 𝑝𝑋)
11103anassrs 1287 . . . . . . . . . 10 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝑝𝑋)
12 psmetsym 22038 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑞𝑋𝑝𝑋) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
137, 9, 11, 12syl3anc 1323 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
14 df-ov 6613 . . . . . . . . 9 (𝑞𝐷𝑝) = (𝐷‘⟨𝑞, 𝑝⟩)
15 df-ov 6613 . . . . . . . . 9 (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
1613, 14, 153eqtr3g 2678 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝐷‘⟨𝑞, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
1716eleq1d 2683 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)))
18 psmetf 22034 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
19 ffun 6010 . . . . . . . . 9 (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
207, 18, 193syl 18 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → Fun 𝐷)
21 simpllr 798 . . . . . . . . . . 11 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝑝𝑋𝑞𝑋))
2221ancomd 467 . . . . . . . . . 10 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝑞𝑋𝑝𝑋))
23 opelxpi 5113 . . . . . . . . . 10 ((𝑞𝑋𝑝𝑋) → ⟨𝑞, 𝑝⟩ ∈ (𝑋 × 𝑋))
2422, 23syl 17 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑞, 𝑝⟩ ∈ (𝑋 × 𝑋))
25 fdm 6013 . . . . . . . . . 10 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
267, 18, 253syl 18 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → dom 𝐷 = (𝑋 × 𝑋))
2724, 26eleqtrrd 2701 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑞, 𝑝⟩ ∈ dom 𝐷)
28 fvimacnv 6293 . . . . . . . 8 ((Fun 𝐷 ∧ ⟨𝑞, 𝑝⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ ⟨𝑞, 𝑝⟩ ∈ (𝐷 “ (0[,)𝑎))))
2920, 27, 28syl2anc 692 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ ⟨𝑞, 𝑝⟩ ∈ (𝐷 “ (0[,)𝑎))))
30 opelxpi 5113 . . . . . . . . . 10 ((𝑝𝑋𝑞𝑋) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
3121, 30syl 17 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
3231, 26eleqtrrd 2701 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
33 fvimacnv 6293 . . . . . . . 8 ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
3420, 32, 33syl2anc 692 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
3517, 29, 343bitr3d 298 . . . . . 6 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ (𝐷 “ (0[,)𝑎)) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
36 simpr 477 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐴 = (𝐷 “ (0[,)𝑎)))
3736eleq2d 2684 . . . . . 6 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑞, 𝑝⟩ ∈ (𝐷 “ (0[,)𝑎))))
3836eleq2d 2684 . . . . . 6 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (⟨𝑝, 𝑞⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
3935, 37, 383bitr4d 300 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
40 eqid 2621 . . . . . . . . 9 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
4140elrnmpt 5337 . . . . . . . 8 (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) → (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4241ibi 256 . . . . . . 7 (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
4342, 1eleq2s 2716 . . . . . 6 (𝐴𝐹 → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
4443ad2antlr 762 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
4539, 44r19.29a 3072 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
46 df-br 4619 . . . . 5 (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
47 vex 3192 . . . . . 6 𝑝 ∈ V
48 vex 3192 . . . . . 6 𝑞 ∈ V
4947, 48opelcnv 5269 . . . . 5 (⟨𝑝, 𝑞⟩ ∈ 𝐴 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
5046, 49bitri 264 . . . 4 (𝑝𝐴𝑞 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
51 df-br 4619 . . . 4 (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
5245, 50, 513bitr4g 303 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ (𝑝𝑋𝑞𝑋)) → (𝑝𝐴𝑞𝑝𝐴𝑞))
53523impb 1257 . 2 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑝𝑋𝑞𝑋) → (𝑝𝐴𝑞𝑝𝐴𝑞))
546, 2, 53eqbrrdva 5256 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  wss 3559  cop 4159   class class class wbr 4618  cmpt 4678   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  Fun wfun 5846  wf 5848  cfv 5852  (class class class)co 6610  0cc0 9888  *cxr 10025  +crp 11784  [,)cico 12127  PsMetcpsmet 19662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-xadd 11899  df-psmet 19670
This theorem is referenced by:  metust  22286
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