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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 477 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
2 | cnvexg 7442 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
3 | 2 | adantl 474 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → ◡𝐴 ∈ V) |
4 | cnvf1o 7612 | . . 3 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) | |
5 | 4 | adantr 473 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
6 | f1oen2g 8321 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ◡𝐴 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) → 𝐴 ≈ ◡𝐴) | |
7 | 1, 3, 5, 6 | syl3anc 1351 | 1 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2050 Vcvv 3409 {csn 4435 ∪ cuni 4708 class class class wbr 4925 ↦ cmpt 5004 ◡ccnv 5402 Rel wrel 5408 –1-1-onto→wf1o 6184 ≈ cen 8301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-1st 7499 df-2nd 7500 df-en 8305 |
This theorem is referenced by: cnvct 8381 cnvfi 8599 lgsquadlem3 25672 |
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