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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 264 | . 2 ⊢ (𝑥 = 𝑋 → (⊤ ↔ ⊤)) | |
4 | ssidd 3990 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑋) | |
5 | elex 3512 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | trud 1547 | . 2 ⊢ (𝑋 ∈ 𝑉 → ⊤) | |
7 | 1, 2, 3, 4, 5, 6 | clcnvlem 40003 | 1 ⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 {cab 2799 ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 ∩ cint 4876 ◡ccnv 5554 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: (None) |
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