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Theorem cnvf1o 7236
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 2621 . 2 (𝑥𝐴 {𝑥}) = (𝑥𝐴 {𝑥})
2 snex 4879 . . . . 5 {𝑥} ∈ V
32cnvex 7075 . . . 4 {𝑥} ∈ V
43uniex 6918 . . 3 {𝑥} ∈ V
54a1i 11 . 2 ((Rel 𝐴𝑥𝐴) → {𝑥} ∈ V)
6 snex 4879 . . . . 5 {𝑦} ∈ V
76cnvex 7075 . . . 4 {𝑦} ∈ V
87uniex 6918 . . 3 {𝑦} ∈ V
98a1i 11 . 2 ((Rel 𝐴𝑦𝐴) → {𝑦} ∈ V)
10 cnvf1olem 7235 . . 3 ((Rel 𝐴 ∧ (𝑥𝐴𝑦 = {𝑥})) → (𝑦𝐴𝑥 = {𝑦}))
11 relcnv 5472 . . . . 5 Rel 𝐴
12 simpr 477 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑦𝐴𝑥 = {𝑦}))
13 cnvf1olem 7235 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
1411, 12, 13sylancr 694 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
15 dfrel2 5552 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
16 eleq2 2687 . . . . . . 7 (𝐴 = 𝐴 → (𝑥𝐴𝑥𝐴))
1715, 16sylbi 207 . . . . . 6 (Rel 𝐴 → (𝑥𝐴𝑥𝐴))
1817anbi1d 740 . . . . 5 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
1918adantr 481 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2014, 19mpbid 222 . . 3 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
2110, 20impbida 876 . 2 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑦𝐴𝑥 = {𝑦})))
221, 5, 9, 21f1od 6850 1 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  {csn 4155   cuni 4409  cmpt 4683  ccnv 5083  Rel wrel 5089  1-1-ontowf1o 5856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-1st 7128  df-2nd 7129
This theorem is referenced by:  tposf12  7337  cnven  7992  xpcomf1o  8009  fsumcnv  14451  fprodcnv  14657  gsumcom2  18314
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