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Mirrors > Home > MPE Home > Th. List > prdsbasmpt2 | Structured version Visualization version GIF version |
Description: A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015.) (Revised by Mario Carneiro, 13-Sep-2015.) |
Ref | Expression |
---|---|
prdsbasmpt2.y | ⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) |
prdsbasmpt2.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsbasmpt2.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsbasmpt2.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsbasmpt2.r | ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) |
prdsbasmpt2.k | ⊢ 𝐾 = (Base‘𝑅) |
Ref | Expression |
---|---|
prdsbasmpt2 | ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbasmpt2.y | . . . 4 ⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) | |
2 | prdsbasmpt2.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
3 | prdsbasmpt2.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsbasmpt2.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | prdsbasmpt2.r | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) | |
6 | prdsbasmpt2.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | prdsbas3 17460 | . . 3 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝐾) |
8 | 7 | eleq2d 2811 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵 ↔ (𝑥 ∈ 𝐼 ↦ 𝑈) ∈ X𝑥 ∈ 𝐼 𝐾)) |
9 | mptelixpg 8950 | . . 3 ⊢ (𝐼 ∈ 𝑊 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ 𝐾)) | |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ 𝐾)) |
11 | 8, 10 | bitrd 278 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ↦ cmpt 5226 ‘cfv 6542 (class class class)co 7415 Xcixp 8912 Basecbs 17177 Xscprds 17424 |
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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 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-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-prds 17426 |
This theorem is referenced by: prdsbascl 17462 prdsgsum 19938 |
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