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Mirrors > Home > MPE Home > Th. List > prdsbascl | Structured version Visualization version GIF version |
Description: An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-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) |
prdsbascl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
prdsbascl | ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbasmpt2.y | . . . . 5 ⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) | |
2 | prdsbasmpt2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
3 | prdsbasmpt2.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsbasmpt2.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | prdsbasmpt2.r | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) | |
6 | eqid 2726 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑥 ∈ 𝐼 ↦ 𝑅) | |
7 | 6 | fnmpt 6703 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 → (𝑥 ∈ 𝐼 ↦ 𝑅) Fn 𝐼) |
8 | 5, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑅) Fn 𝐼) |
9 | prdsbascl.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
10 | 1, 2, 3, 4, 8, 9 | prdsbasfn 17488 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
11 | dffn5 6963 | . . . 4 ⊢ (𝐹 Fn 𝐼 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) | |
12 | 10, 11 | sylib 217 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
13 | 12, 9 | eqeltrrd 2827 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ 𝐵) |
14 | prdsbasmpt2.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
15 | 1, 2, 3, 4, 5, 14 | prdsbasmpt2 17499 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐾)) |
16 | 13, 15 | mpbid 231 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ↦ cmpt 5238 Fn wfn 6551 ‘cfv 6556 (class class class)co 7426 Basecbs 17215 Xscprds 17462 |
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 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9487 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-fz 13541 df-struct 17151 df-slot 17186 df-ndx 17198 df-base 17216 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-ip 17286 df-tset 17287 df-ple 17288 df-ds 17290 df-hom 17292 df-cco 17293 df-prds 17464 |
This theorem is referenced by: prdsdsval3 17502 prdsdsf 24367 prdsxmetlem 24368 prdsmet 24370 prdsbl 24494 prdsxmslem2 24532 prdsbnd 37496 prdsbnd2 37498 rrnequiv 37538 |
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