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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmtrcl | Structured version Visualization version GIF version |
Description: The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.) |
Ref | Expression |
---|---|
dmtrcl | ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trclubg 14361 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) | |
2 | dmss 5773 | . . . 4 ⊢ (∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋))) |
4 | dmun 5781 | . . . 4 ⊢ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) | |
5 | dmxpss 6030 | . . . . 5 ⊢ dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋 | |
6 | ssequn2 4161 | . . . . 5 ⊢ (dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋 ↔ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋) | |
7 | 5, 6 | mpbi 232 | . . . 4 ⊢ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋 |
8 | 4, 7 | eqtri 2846 | . . 3 ⊢ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = dom 𝑋 |
9 | 3, 8 | sseqtrdi 4019 | . 2 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ dom 𝑋) |
10 | ssmin 4897 | . . 3 ⊢ 𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} | |
11 | dmss 5773 | . . 3 ⊢ (𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} → dom 𝑋 ⊆ dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}) | |
12 | 10, 11 | mp1i 13 | . 2 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ⊆ dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}) |
13 | 9, 12 | eqssd 3986 | 1 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 ∪ cun 3936 ⊆ wss 3938 ∩ cint 4878 × cxp 5555 dom cdm 5557 ran crn 5558 ∘ ccom 5561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 |
This theorem is referenced by: dfrtrcl5 39996 |
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