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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmptiunrelexplb0d | Structured version Visualization version GIF version | ||
| Description: If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.) |
| Ref | Expression |
|---|---|
| fvmptiunrelexplb0d.c | ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) |
| fvmptiunrelexplb0d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
| fvmptiunrelexplb0d.n | ⊢ (𝜑 → 𝑁 ∈ V) |
| fvmptiunrelexplb0d.0 | ⊢ (𝜑 → 0 ∈ 𝑁) |
| Ref | Expression |
|---|---|
| fvmptiunrelexplb0d | ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvmptiunrelexplb0d.0 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑁) | |
| 2 | oveq2 7439 | . . . 4 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0)) | |
| 3 | 2 | ssiun2s 5048 | . . 3 ⊢ (0 ∈ 𝑁 → (𝑅↑𝑟0) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
| 4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → (𝑅↑𝑟0) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
| 5 | fvmptiunrelexplb0d.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
| 6 | relexp0g 15061 | . . 3 ⊢ (𝑅 ∈ V → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | |
| 7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 8 | fvmptiunrelexplb0d.c | . . . 4 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) | |
| 9 | oveq1 7438 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
| 10 | 9 | iuneq2d 5022 | . . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
| 11 | fvmptiunrelexplb0d.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ V) | |
| 12 | ovex 7464 | . . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
| 13 | 12 | rgenw 3065 | . . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V |
| 14 | iunexg 7988 | . . . . 5 ⊢ ((𝑁 ∈ V ∧ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) | |
| 15 | 11, 13, 14 | sylancl 586 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) |
| 16 | 8, 10, 5, 15 | fvmptd3 7039 | . . 3 ⊢ (𝜑 → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
| 17 | 16 | eqcomd 2743 | . 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅)) |
| 18 | 4, 7, 17 | 3sstr3d 4038 | 1 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 ∪ ciun 4991 ↦ cmpt 5225 I cid 5577 dom cdm 5685 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ↑𝑟crelexp 15058 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 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-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-n0 12527 df-relexp 15059 |
| This theorem is referenced by: fvmptiunrelexplb0da 43698 fvrcllb0d 43706 fvrtrcllb0d 43748 |
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