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Mirrors > Home > MPE Home > Th. List > cnven | Structured version Visualization version 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 485 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
2 | cnvexg 7771 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
3 | 2 | adantl 482 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → ◡𝐴 ∈ V) |
4 | cnvf1o 7951 | . . 3 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) | |
5 | 4 | adantr 481 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
6 | f1oen2g 8756 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ◡𝐴 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) → 𝐴 ≈ ◡𝐴) | |
7 | 1, 3, 5, 6 | syl3anc 1370 | 1 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3432 {csn 4561 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 ◡ccnv 5588 Rel wrel 5594 –1-1-onto→wf1o 6432 ≈ cen 8730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-1st 7831 df-2nd 7832 df-en 8734 |
This theorem is referenced by: cnvct 8824 cnvfiALT 9101 lgsquadlem3 26530 |
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