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Theorem cnvf1o 5871
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 2054 . 2 (𝑥𝐴 {𝑥}) = (𝑥𝐴 {𝑥})
2 snexg 3961 . . . 4 (𝑥𝐴 → {𝑥} ∈ V)
3 cnvexg 4880 . . . 4 ({𝑥} ∈ V → {𝑥} ∈ V)
4 uniexg 4200 . . . 4 ({𝑥} ∈ V → {𝑥} ∈ V)
52, 3, 43syl 17 . . 3 (𝑥𝐴 {𝑥} ∈ V)
65adantl 266 . 2 ((Rel 𝐴𝑥𝐴) → {𝑥} ∈ V)
7 snexg 3961 . . . 4 (𝑦𝐴 → {𝑦} ∈ V)
8 cnvexg 4880 . . . 4 ({𝑦} ∈ V → {𝑦} ∈ V)
9 uniexg 4200 . . . 4 ({𝑦} ∈ V → {𝑦} ∈ V)
107, 8, 93syl 17 . . 3 (𝑦𝐴 {𝑦} ∈ V)
1110adantl 266 . 2 ((Rel 𝐴𝑦𝐴) → {𝑦} ∈ V)
12 cnvf1olem 5870 . . 3 ((Rel 𝐴 ∧ (𝑥𝐴𝑦 = {𝑥})) → (𝑦𝐴𝑥 = {𝑦}))
13 relcnv 4728 . . . . 5 Rel 𝐴
14 simpr 107 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑦𝐴𝑥 = {𝑦}))
15 cnvf1olem 5870 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
1613, 14, 15sylancr 399 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
17 dfrel2 4796 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
18 eleq2 2115 . . . . . . 7 (𝐴 = 𝐴 → (𝑥𝐴𝑥𝐴))
1917, 18sylbi 118 . . . . . 6 (Rel 𝐴 → (𝑥𝐴𝑥𝐴))
2019anbi1d 446 . . . . 5 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2120adantr 265 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2216, 21mpbid 139 . . 3 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
2312, 22impbida 536 . 2 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑦𝐴𝑥 = {𝑦})))
241, 6, 11, 23f1od 5728 1 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wcel 1407  Vcvv 2572  {csn 3400   cuni 3605  cmpt 3843  ccnv 4369  Rel wrel 4375  1-1-ontowf1o 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-1st 5792  df-2nd 5793
This theorem is referenced by:  tposf12  5912  cnven  6316  xpcomf1o  6327
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