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Mirrors > Home > MPE Home > Th. List > pwselbas | Structured version Visualization version GIF version |
Description: An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.) |
Ref | Expression |
---|---|
pwsbas.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsbas.f | ⊢ 𝐵 = (Base‘𝑅) |
pwselbas.v | ⊢ 𝑉 = (Base‘𝑌) |
pwselbas.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
pwselbas.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
pwselbas.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
pwselbas | ⊢ (𝜑 → 𝑋:𝐼⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwselbas.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | pwselbas.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
3 | pwselbas.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
4 | pwsbas.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
5 | pwsbas.f | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
6 | pwselbas.v | . . . 4 ⊢ 𝑉 = (Base‘𝑌) | |
7 | 4, 5, 6 | pwselbasb 17463 | . . 3 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍) → (𝑋 ∈ 𝑉 ↔ 𝑋:𝐼⟶𝐵)) |
8 | 2, 3, 7 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑉 ↔ 𝑋:𝐼⟶𝐵)) |
9 | 1, 8 | mpbid 231 | 1 ⊢ (𝜑 → 𝑋:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 ↑s cpws 17421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-prds 17422 df-pws 17424 |
This theorem is referenced by: pwsplusgval 17465 pwsmulrval 17466 pwsle 17467 pwsleval 17468 pwsvscafval 17469 pwsvscaval 17470 pwsco1mhm 18777 pwsco2mhm 18778 pwsinvg 19002 pwssub 19003 pwspjmhmmgpd 20257 mpff 22043 fveval1fvcl 22245 evl1addd 22253 evl1subd 22254 evl1muld 22255 pf1f 22262 pf1mpf 22264 ply1remlem 26092 ply1rem 26093 fta1glem1 26095 fta1glem2 26096 fta1g 26097 fta1blem 26098 idomrootle 26100 plypf1 26139 lgsqrlem2 27273 lgsqrlem3 27274 evls1fvf 33231 evls1fpws 33240 elirng 33354 irngss 33355 irngnzply1lem 33358 irngnzply1 33359 pwsgprod 41768 evlscl 41785 evlsvvval 41790 evlsaddval 41795 evlsmulval 41796 evlcl 41799 evladdval 41802 evlmulval 41803 selvcl 41810 pwssplit4 42507 |
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