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Mirrors > Home > MPE Home > Th. List > Mathboxes > rntrclfvOAI | Structured version Visualization version GIF version |
Description: The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.) |
Ref | Expression |
---|---|
rntrclfvOAI | ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trclfvub 13596 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | |
2 | rnss 5275 | . . . 4 ⊢ ((t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅))) |
4 | rnun 5460 | . . . . 5 ⊢ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅))) |
6 | rnxpss 5485 | . . . . 5 ⊢ ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 | |
7 | ssequn2 3748 | . . . . 5 ⊢ (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅) | |
8 | 6, 7 | mpbi 219 | . . . 4 ⊢ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅 |
9 | 5, 8 | syl6eq 2660 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅) |
10 | 3, 9 | sseqtrd 3604 | . 2 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅) |
11 | trclfvlb 13597 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) | |
12 | rnss 5275 | . . 3 ⊢ (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅)) |
14 | 10, 13 | eqssd 3585 | 1 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 × cxp 5036 dom cdm 5038 ran crn 5039 ‘cfv 5804 t+ctcl 13572 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-rab 2905 df-v 3175 df-sbc 3403 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-int 4411 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-iota 5768 df-fun 5806 df-fv 5812 df-trcl 13574 |
This theorem is referenced by: (None) |
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