Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cnven | GIF version |
Description: A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
cnven | ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 108 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
2 | cnvexg 4875 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
3 | 2 | adantl 271 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → ◡𝐴 ∈ V) |
4 | cnvf1o 5866 | . . 3 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) | |
5 | 4 | adantr 270 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
6 | f1oen2g 6258 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ◡𝐴 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) → 𝐴 ≈ ◡𝐴) | |
7 | 1, 3, 5, 6 | syl3anc 1169 | 1 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 Vcvv 2601 {csn 3398 ∪ cuni 3601 class class class wbr 3785 ↦ cmpt 3839 ◡ccnv 4362 Rel wrel 4368 –1-1-onto→wf1o 4921 ≈ cen 6242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1st 5787 df-2nd 5788 df-en 6245 |
This theorem is referenced by: relcnvfi 6391 |
Copyright terms: Public domain | W3C validator |