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Theorem cnvf1o 8136
Description: Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
cnvf1o (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem cnvf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . 2 (𝑥𝐴 {𝑥}) = (𝑥𝐴 {𝑥})
2 vsnex 5434 . . . . 5 {𝑥} ∈ V
32cnvex 7947 . . . 4 {𝑥} ∈ V
43uniex 7761 . . 3 {𝑥} ∈ V
54a1i 11 . 2 ((Rel 𝐴𝑥𝐴) → {𝑥} ∈ V)
6 vsnex 5434 . . . . 5 {𝑦} ∈ V
76cnvex 7947 . . . 4 {𝑦} ∈ V
87uniex 7761 . . 3 {𝑦} ∈ V
98a1i 11 . 2 ((Rel 𝐴𝑦𝐴) → {𝑦} ∈ V)
10 cnvf1olem 8135 . . 3 ((Rel 𝐴 ∧ (𝑥𝐴𝑦 = {𝑥})) → (𝑦𝐴𝑥 = {𝑦}))
11 relcnv 6122 . . . . 5 Rel 𝐴
12 simpr 484 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑦𝐴𝑥 = {𝑦}))
13 cnvf1olem 8135 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
1411, 12, 13sylancr 587 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
15 dfrel2 6209 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
16 eleq2 2830 . . . . . . 7 (𝐴 = 𝐴 → (𝑥𝐴𝑥𝐴))
1715, 16sylbi 217 . . . . . 6 (Rel 𝐴 → (𝑥𝐴𝑥𝐴))
1817anbi1d 631 . . . . 5 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
1918adantr 480 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2014, 19mpbid 232 . . 3 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
2110, 20impbida 801 . 2 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑦𝐴𝑥 = {𝑦})))
221, 5, 9, 21f1od 7685 1 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626   cuni 4907  cmpt 5225  ccnv 5684  Rel wrel 5690  1-1-ontowf1o 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  tposf12  8276  cnven  9073  xpcomf1o  9101  fsumcnv  15809  fprodcnv  16019  gsumcom2  19993  tposres3  48781
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