![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 5053 | . . 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 15058 | . . 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 5027 | . . . 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 3063 | . . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V |
14 | iunexg 7987 | . . . . 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 2741 | . 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅)) |
18 | 4, 7, 17 | 3sstr3d 4042 | 1 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 ∪ ciun 4996 ↦ cmpt 5231 I cid 5582 dom cdm 5689 ran crn 5690 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ↑𝑟crelexp 15055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-n0 12525 df-relexp 15056 |
This theorem is referenced by: fvmptiunrelexplb0da 43675 fvrcllb0d 43683 fvrtrcllb0d 43725 |
Copyright terms: Public domain | W3C validator |