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Mirrors > Home > MPE Home > Th. List > prdsbasprj | Structured version Visualization version GIF version |
Description: Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsbasmpt.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsbasmpt.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsbasmpt.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsbasmpt.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsbasmpt.r | ⊢ (𝜑 → 𝑅 Fn 𝐼) |
prdsbasmpt.t | ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
prdsbasprj.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
Ref | Expression |
---|---|
prdsbasprj | ⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . 3 ⊢ (𝑥 = 𝐽 → (𝑇‘𝑥) = (𝑇‘𝐽)) | |
2 | 2fveq3 6901 | . . 3 ⊢ (𝑥 = 𝐽 → (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝐽))) | |
3 | 1, 2 | eleq12d 2819 | . 2 ⊢ (𝑥 = 𝐽 → ((𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥)) ↔ (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽)))) |
4 | prdsbasmpt.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐵) | |
5 | prdsbasmpt.y | . . . . 5 ⊢ 𝑌 = (𝑆Xs𝑅) | |
6 | prdsbasmpt.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
7 | prdsbasmpt.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
8 | prdsbasmpt.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
9 | prdsbasmpt.r | . . . . 5 ⊢ (𝜑 → 𝑅 Fn 𝐼) | |
10 | 5, 6, 7, 8, 9 | prdsbas2 17454 | . . . 4 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
11 | 4, 10 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
12 | elixp2 8920 | . . . 4 ⊢ (𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↔ (𝑇 ∈ V ∧ 𝑇 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥)))) | |
13 | 12 | simp3bi 1144 | . . 3 ⊢ (𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) → ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥))) |
14 | 11, 13 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥))) |
15 | prdsbasprj.j | . 2 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
16 | 3, 14, 15 | rspcdva 3607 | 1 ⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 Xcixp 8916 Basecbs 17183 Xscprds 17430 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-hom 17260 df-cco 17261 df-prds 17432 |
This theorem is referenced by: prdsplusgsgrpcl 18695 prdssgrpd 18696 prdsplusgcl 18728 prdsidlem 18729 prdsmndd 18730 prdspjmhm 18789 prdsinvlem 19013 prdscmnd 19828 prdsmulrngcl 20127 prdsrngd 20128 prdsringd 20269 prdsvscacl 20864 prdslmodd 20865 |
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