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| Mirrors > Home > MPE Home > Th. List > prdsvallem | Structured version Visualization version GIF version | ||
| Description: Lemma for prdsval 17500. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17500, 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 3484 | . 2 ⊢ 𝑣 ∈ V | |
| 2 | ovex 7464 | . . 3 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V | |
| 3 | 2 | pwex 5380 | . 2 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V |
| 4 | ovssunirn 7467 | . . . . . . . 8 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥)) | |
| 5 | homid 17456 | . . . . . . . . . . 11 ⊢ Hom = Slot (Hom ‘ndx) | |
| 6 | 5 | strfvss 17224 | . . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥) |
| 7 | fvssunirn 6939 | . . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟 | |
| 8 | rnss 5950 | . . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟) | |
| 9 | uniss 4915 | . . . . . . . . . . 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 3993 | . . . . . . . . 9 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 |
| 12 | rnss 5950 | . . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟) | |
| 13 | uniss 4915 | . . . . . . . . 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 3993 | . . . . . . 7 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 |
| 16 | 15 | rgenw 3065 | . . . . . 6 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 |
| 17 | ss2ixp 8950 | . . . . . 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 3484 | . . . . . . 7 ⊢ 𝑟 ∈ V | |
| 20 | 19 | dmex 7931 | . . . . . 6 ⊢ dom 𝑟 ∈ V |
| 21 | 19 | rnex 7932 | . . . . . . . . . . 11 ⊢ ran 𝑟 ∈ V |
| 22 | 21 | uniex 7761 | . . . . . . . . . 10 ⊢ ∪ ran 𝑟 ∈ V |
| 23 | 22 | rnex 7932 | . . . . . . . . 9 ⊢ ran ∪ ran 𝑟 ∈ V |
| 24 | 23 | uniex 7761 | . . . . . . . 8 ⊢ ∪ ran ∪ ran 𝑟 ∈ V |
| 25 | 24 | rnex 7932 | . . . . . . 7 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V |
| 26 | 25 | uniex 7761 | . . . . . 6 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V |
| 27 | 20, 26 | ixpconst 8947 | . . . . 5 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) |
| 28 | 18, 27 | sseqtri 4032 | . . . 4 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) |
| 29 | 2, 28 | elpwi2 5335 | . . 3 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) |
| 30 | 29 | rgen2w 3066 | . 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 8103 | 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 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 dom cdm 5685 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8866 Xcixp 8937 ndxcnx 17230 Hom chom 17308 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 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-dec 12734 df-slot 17219 df-ndx 17231 df-hom 17321 |
| This theorem is referenced by: prdsval 17500 |
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