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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 14361 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | |
2 | rnss 5803 | . . . 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 5998 | . . . . 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 6023 | . . . . 5 ⊢ ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 | |
7 | ssequn2 4158 | . . . . 5 ⊢ (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅) | |
8 | 6, 7 | mpbi 232 | . . . 4 ⊢ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅 |
9 | 5, 8 | syl6eq 2872 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅) |
10 | 3, 9 | sseqtrd 4006 | . 2 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅) |
11 | trclfvlb 14362 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) | |
12 | rnss 5803 | . . 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 3983 | 1 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 ⊆ wss 3935 × cxp 5547 dom cdm 5549 ran crn 5550 ‘cfv 6349 t+ctcl 14339 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-trcl 14341 |
This theorem is referenced by: (None) |
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