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Mirrors > Home > MPE Home > Th. List > prdsvallem | Structured version Visualization version GIF version |
Description: Lemma for prdsval 17501. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17501, dependency on df-hom 17321 removed. (Revised by AV, 13-Oct-2024.) |
Ref | Expression |
---|---|
prdsvallem | ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3481 | . 2 ⊢ 𝑣 ∈ V | |
2 | ovex 7463 | . . 3 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V | |
3 | 2 | pwex 5385 | . 2 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V |
4 | ovssunirn 7466 | . . . . . . . 8 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥)) | |
5 | homid 17457 | . . . . . . . . . . 11 ⊢ Hom = Slot (Hom ‘ndx) | |
6 | 5 | strfvss 17220 | . . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥) |
7 | fvssunirn 6939 | . . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟 | |
8 | rnss 5952 | . . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟) | |
9 | uniss 4919 | . . . . . . . . . . 11 ⊢ (ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟 → ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟) | |
10 | 7, 8, 9 | mp2b 10 | . . . . . . . . . 10 ⊢ ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟 |
11 | 6, 10 | sstri 4004 | . . . . . . . . 9 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 |
12 | rnss 5952 | . . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟) | |
13 | uniss 4919 | . . . . . . . . 9 ⊢ (ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟 → ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟) | |
14 | 11, 12, 13 | mp2b 10 | . . . . . . . 8 ⊢ ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 |
15 | 4, 14 | sstri 4004 | . . . . . . 7 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 |
16 | 15 | rgenw 3062 | . . . . . 6 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 |
17 | ss2ixp 8948 | . . . . . 6 ⊢ (∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 → X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟) | |
18 | 16, 17 | ax-mp 5 | . . . . 5 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 |
19 | vex 3481 | . . . . . . 7 ⊢ 𝑟 ∈ V | |
20 | 19 | dmex 7931 | . . . . . 6 ⊢ dom 𝑟 ∈ V |
21 | 19 | rnex 7932 | . . . . . . . . . . 11 ⊢ ran 𝑟 ∈ V |
22 | 21 | uniex 7759 | . . . . . . . . . 10 ⊢ ∪ ran 𝑟 ∈ V |
23 | 22 | rnex 7932 | . . . . . . . . 9 ⊢ ran ∪ ran 𝑟 ∈ V |
24 | 23 | uniex 7759 | . . . . . . . 8 ⊢ ∪ ran ∪ ran 𝑟 ∈ V |
25 | 24 | rnex 7932 | . . . . . . 7 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V |
26 | 25 | uniex 7759 | . . . . . 6 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V |
27 | 20, 26 | ixpconst 8945 | . . . . 5 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) |
28 | 18, 27 | sseqtri 4031 | . . . 4 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) |
29 | 2, 28 | elpwi2 5340 | . . 3 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) |
30 | 29 | rgen2w 3063 | . 2 ⊢ ∀𝑓 ∈ 𝑣 ∀𝑔 ∈ 𝑣 X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) |
31 | 1, 1, 3, 30 | mpoexw 8101 | 1 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ⊆ wss 3962 𝒫 cpw 4604 ∪ cuni 4911 dom cdm 5688 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ↑m cmap 8864 Xcixp 8935 ndxcnx 17226 Hom chom 17308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 df-slot 17215 df-ndx 17227 df-hom 17321 |
This theorem is referenced by: prdsval 17501 |
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