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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 6673 | . . 3 ⊢ (𝑥 = 𝐽 → (𝑇‘𝑥) = (𝑇‘𝐽)) | |
2 | 2fveq3 6678 | . . 3 ⊢ (𝑥 = 𝐽 → (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝐽))) | |
3 | 1, 2 | eleq12d 2910 | . 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 16745 | . . . 4 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
11 | 4, 10 | eleqtrd 2918 | . . 3 ⊢ (𝜑 → 𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
12 | elixp2 8468 | . . . 4 ⊢ (𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↔ (𝑇 ∈ V ∧ 𝑇 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥)))) | |
13 | 12 | simp3bi 1143 | . . 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 3628 | 1 ⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 Fn wfn 6353 ‘cfv 6358 (class class class)co 7159 Xcixp 8464 Basecbs 16486 Xscprds 16722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-hom 16592 df-cco 16593 df-prds 16724 |
This theorem is referenced by: prdsplusgcl 17945 prdsidlem 17946 prdsmndd 17947 prdspjmhm 17996 prdsinvlem 18211 prdscmnd 18984 prdsmulrcl 19364 prdsringd 19365 prdsvscacl 19743 prdslmodd 19744 |
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