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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 6663 | . . 3 ⊢ (𝑥 = 𝐽 → (𝑇‘𝑥) = (𝑇‘𝐽)) | |
2 | 2fveq3 6668 | . . 3 ⊢ (𝑥 = 𝐽 → (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝐽))) | |
3 | 1, 2 | eleq12d 2905 | . 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 16734 | . . . 4 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
11 | 4, 10 | eleqtrd 2913 | . . 3 ⊢ (𝜑 → 𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
12 | elixp2 8457 | . . . 4 ⊢ (𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↔ (𝑇 ∈ V ∧ 𝑇 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥)))) | |
13 | 12 | simp3bi 1142 | . . 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 3623 | 1 ⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ∀wral 3136 Vcvv 3493 Fn wfn 6343 ‘cfv 6348 (class class class)co 7148 Xcixp 8453 Basecbs 16475 Xscprds 16711 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-prds 16713 |
This theorem is referenced by: prdsplusgcl 17934 prdsidlem 17935 prdsmndd 17936 prdspjmhm 17985 prdsinvlem 18200 prdscmnd 18973 prdsmulrcl 19353 prdsringd 19354 prdsvscacl 19732 prdslmodd 19733 |
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