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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvtrucl0 | Structured version Visualization version GIF version |
Description: The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
Ref | Expression |
---|---|
cnvtrucl0 | ⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idd 24 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (⊤ → ⊤)) | |
2 | idd 24 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ◡𝑥) → (⊤ → ⊤)) | |
3 | biidd 251 | . 2 ⊢ (𝑥 = 𝑋 → (⊤ ↔ ⊤)) | |
4 | ssid 3587 | . . 3 ⊢ 𝑋 ⊆ 𝑋 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑋) |
6 | elex 3185 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
7 | a1tru 1491 | . 2 ⊢ (𝑋 ∈ 𝑉 → ⊤) | |
8 | 1, 2, 3, 5, 6, 7 | clcnvlem 36949 | 1 ⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {cab 2596 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∩ cint 4410 ◡ccnv 5037 |
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-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-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: (None) |
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