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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 6557 | . . . 4 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0)) | |
3 | 2 | ssiun2s 4500 | . . 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 13610 | . . 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.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ V) | |
9 | ovex 6577 | . . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
10 | 9 | rgenw 2908 | . . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V |
11 | iunexg 7035 | . . . . 5 ⊢ ((𝑁 ∈ V ∧ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) | |
12 | 8, 10, 11 | sylancl 693 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) |
13 | oveq1 6556 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
14 | 13 | iuneq2d 4483 | . . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
15 | fvmptiunrelexplb0d.c | . . . . 5 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) | |
16 | 14, 15 | fvmptg 6189 | . . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
17 | 5, 12, 16 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
18 | 17 | eqcomd 2616 | . 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅)) |
19 | 4, 7, 18 | 3sstr3d 3610 | 1 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 ∪ ciun 4455 ↦ cmpt 4643 I cid 4948 dom cdm 5038 ran crn 5039 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ↑𝑟crelexp 13608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-n0 11170 df-relexp 13609 |
This theorem is referenced by: fvmptiunrelexplb0da 36996 fvrcllb0d 37004 fvrtrcllb0d 37046 |
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