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Theorem cnvf1o 7478
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 2765 . 2 (𝑥𝐴 {𝑥}) = (𝑥𝐴 {𝑥})
2 snex 5064 . . . . 5 {𝑥} ∈ V
32cnvex 7311 . . . 4 {𝑥} ∈ V
43uniex 7151 . . 3 {𝑥} ∈ V
54a1i 11 . 2 ((Rel 𝐴𝑥𝐴) → {𝑥} ∈ V)
6 snex 5064 . . . . 5 {𝑦} ∈ V
76cnvex 7311 . . . 4 {𝑦} ∈ V
87uniex 7151 . . 3 {𝑦} ∈ V
98a1i 11 . 2 ((Rel 𝐴𝑦𝐴) → {𝑦} ∈ V)
10 cnvf1olem 7477 . . 3 ((Rel 𝐴 ∧ (𝑥𝐴𝑦 = {𝑥})) → (𝑦𝐴𝑥 = {𝑦}))
11 relcnv 5685 . . . . 5 Rel 𝐴
12 simpr 477 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑦𝐴𝑥 = {𝑦}))
13 cnvf1olem 7477 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
1411, 12, 13sylancr 581 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
15 dfrel2 5766 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
16 eleq2 2833 . . . . . . 7 (𝐴 = 𝐴 → (𝑥𝐴𝑥𝐴))
1715, 16sylbi 208 . . . . . 6 (Rel 𝐴 → (𝑥𝐴𝑥𝐴))
1817anbi1d 623 . . . . 5 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
1918adantr 472 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2014, 19mpbid 223 . . 3 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
2110, 20impbida 835 . 2 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑦𝐴𝑥 = {𝑦})))
221, 5, 9, 21f1od 7083 1 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  {csn 4334   cuni 4594  cmpt 4888  ccnv 5276  Rel wrel 5282  1-1-ontowf1o 6067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-1st 7366  df-2nd 7367
This theorem is referenced by:  tposf12  7580  cnven  8236  xpcomf1o  8256  fsumcnv  14789  fprodcnv  14996  gsumcom2  18640
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