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Mirrors > Home > MPE Home > Th. List > cnvf1o | Structured version Visualization version GIF version |
Description: Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
cnvf1o | ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) | |
2 | vsnex 5429 | . . . . 5 ⊢ {𝑥} ∈ V | |
3 | 2 | cnvex 7920 | . . . 4 ⊢ ◡{𝑥} ∈ V |
4 | 3 | uniex 7735 | . . 3 ⊢ ∪ ◡{𝑥} ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∪ ◡{𝑥} ∈ V) |
6 | vsnex 5429 | . . . . 5 ⊢ {𝑦} ∈ V | |
7 | 6 | cnvex 7920 | . . . 4 ⊢ ◡{𝑦} ∈ V |
8 | 7 | uniex 7735 | . . 3 ⊢ ∪ ◡{𝑦} ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ ◡𝐴) → ∪ ◡{𝑦} ∈ V) |
10 | cnvf1olem 8101 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) | |
11 | relcnv 6103 | . . . . 5 ⊢ Rel ◡𝐴 | |
12 | simpr 484 | . . . . 5 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) | |
13 | cnvf1olem 8101 | . . . . 5 ⊢ ((Rel ◡𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) | |
14 | 11, 12, 13 | sylancr 586 | . . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) |
15 | dfrel2 6188 | . . . . . . 7 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
16 | eleq2 2821 | . . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴)) | |
17 | 15, 16 | sylbi 216 | . . . . . 6 ⊢ (Rel 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴)) |
18 | 17 | anbi1d 629 | . . . . 5 ⊢ (Rel 𝐴 → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))) |
20 | 14, 19 | mpbid 231 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) |
21 | 10, 20 | impbida 798 | . 2 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))) |
22 | 1, 5, 9, 21 | f1od 7662 | 1 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {csn 4628 ∪ cuni 4908 ↦ cmpt 5231 ◡ccnv 5675 Rel wrel 5681 –1-1-onto→wf1o 6542 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1st 7979 df-2nd 7980 |
This theorem is referenced by: tposf12 8242 cnven 9039 xpcomf1o 9067 fsumcnv 15726 fprodcnv 15934 gsumcom2 19891 |
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