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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 3988 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑋) | |
5 | elex 3511 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | trud 1541 | . 2 ⊢ (𝑋 ∈ 𝑉 → ⊤) | |
7 | 1, 2, 3, 4, 5, 6 | clcnvlem 39974 | 1 ⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ⊤wtru 1532 ∈ wcel 2108 {cab 2797 ∖ cdif 3931 ∪ cun 3932 ⊆ wss 3934 ∩ cint 4867 ◡ccnv 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7681 df-2nd 7682 |
This theorem is referenced by: (None) |
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