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Theorem cnvf1o 7797
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 2826 . 2 (𝑥𝐴 {𝑥}) = (𝑥𝐴 {𝑥})
2 snex 5328 . . . . 5 {𝑥} ∈ V
32cnvex 7618 . . . 4 {𝑥} ∈ V
43uniex 7455 . . 3 {𝑥} ∈ V
54a1i 11 . 2 ((Rel 𝐴𝑥𝐴) → {𝑥} ∈ V)
6 snex 5328 . . . . 5 {𝑦} ∈ V
76cnvex 7618 . . . 4 {𝑦} ∈ V
87uniex 7455 . . 3 {𝑦} ∈ V
98a1i 11 . 2 ((Rel 𝐴𝑦𝐴) → {𝑦} ∈ V)
10 cnvf1olem 7796 . . 3 ((Rel 𝐴 ∧ (𝑥𝐴𝑦 = {𝑥})) → (𝑦𝐴𝑥 = {𝑦}))
11 relcnv 5965 . . . . 5 Rel 𝐴
12 simpr 485 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑦𝐴𝑥 = {𝑦}))
13 cnvf1olem 7796 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
1411, 12, 13sylancr 587 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
15 dfrel2 6044 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
16 eleq2 2906 . . . . . . 7 (𝐴 = 𝐴 → (𝑥𝐴𝑥𝐴))
1715, 16sylbi 218 . . . . . 6 (Rel 𝐴 → (𝑥𝐴𝑥𝐴))
1817anbi1d 629 . . . . 5 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
1918adantr 481 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2014, 19mpbid 233 . . 3 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
2110, 20impbida 797 . 2 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑦𝐴𝑥 = {𝑦})))
221, 5, 9, 21f1od 7387 1 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  {csn 4564   cuni 4837  cmpt 5143  ccnv 5553  Rel wrel 5559  1-1-ontowf1o 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-1st 7680  df-2nd 7681
This theorem is referenced by:  tposf12  7908  cnven  8574  xpcomf1o  8595  fsumcnv  15118  fprodcnv  15327  gsumcom2  19015
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