fzossfz - Metamath Proof Explorer

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Theorem fzossfz 13681

Description: A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)

**Assertion**
Ref	Expression
fzossfz	⊢ (𝐴..^𝐵) ⊆ (𝐴...𝐵)

Proof of Theorem fzossfz Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzofz 13678	. 2 ⊢ (𝑥 ∈ (𝐴..^𝐵) → 𝑥 ∈ (𝐴...𝐵))
2	1	ssriv 3976	1 ⊢ (𝐴..^𝐵) ⊆ (𝐴...𝐵)

Colors of variables: wff setvar class

Syntax hints: ⊆ wss 3939 (class class class)co 7414 ...cfz 13514 ..^cfzo 13657

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 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658

This theorem is referenced by: fzossz 13682 fzossnn0 13693 fzossnn 13711 elfzom1elp1fzo 13729 injresinjlem 13782 injresinj 13783 zmodfzp1 13890 uzindi 13977 wrdind 14702 wrd2ind 14703 scshwfzeqfzo 14807 telfsumo 15778 dfphi2 16740 cshwshashlem1 17062 psgnunilem5 19451 psgnunilem2 19452 efgredlemf 19698 efgredlemd 19701 efgredlemc 19702 uspgr2wlkeq 29476 wlkres 29500 redwlklem 29501 trlreslem 29529 pthdivtx 29559 eucrct2eupth 30071 cycpmfv2 32852 signstfvn 34230 signsvtn0 34231 breprexplemc 34293 pfxwlk 34762 fzossuz 44798 fourierdlem20 45550 fourierdlem25 45555 fourierdlem37 45567 fourierdlem64 45593 fourierdlem79 45608 fourierdlem89 45618 fourierdlem91 45620 fourierdlem101 45630 iccpartres 46793 iccpartipre 46796 iccpartleu 46803 bgoldbtbndlem2 47181

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