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Mirrors > Home > MPE Home > Th. List > setsmsbas | Structured version Visualization version GIF version |
Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 12-Nov-2024.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
Ref | Expression |
---|---|
setsmsbas | ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 17085 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
2 | tsetndxnbasendx 17236 | . . . 4 ⊢ (TopSet‘ndx) ≠ (Base‘ndx) | |
3 | 2 | necomi 2998 | . . 3 ⊢ (Base‘ndx) ≠ (TopSet‘ndx) |
4 | 1, 3 | setsnid 17080 | . 2 ⊢ (Base‘𝑀) = (Base‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
5 | setsms.x | . 2 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
6 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
7 | 6 | fveq2d 6846 | . 2 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
8 | 4, 5, 7 | 3eqtr4a 2802 | 1 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 〈cop 4592 × cxp 5631 ↾ cres 5635 ‘cfv 6496 (class class class)co 7356 sSet csts 17034 ndxcnx 17064 Basecbs 17082 TopSetcts 17138 distcds 17141 MetOpencmopn 20784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-tset 17151 |
This theorem is referenced by: setsmstopn 23831 setsxms 23832 setsms 23833 tmslem 23835 tmslemOLD 23836 |
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