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Theorem cnvf1o 5846
 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 2040 . 2 (𝑥𝐴 {𝑥}) = (𝑥𝐴 {𝑥})
2 snexg 3936 . . . 4 (𝑥𝐴 → {𝑥} ∈ V)
3 cnvexg 4855 . . . 4 ({𝑥} ∈ V → {𝑥} ∈ V)
4 uniexg 4175 . . . 4 ({𝑥} ∈ V → {𝑥} ∈ V)
52, 3, 43syl 17 . . 3 (𝑥𝐴 {𝑥} ∈ V)
65adantl 262 . 2 ((Rel 𝐴𝑥𝐴) → {𝑥} ∈ V)
7 snexg 3936 . . . 4 (𝑦𝐴 → {𝑦} ∈ V)
8 cnvexg 4855 . . . 4 ({𝑦} ∈ V → {𝑦} ∈ V)
9 uniexg 4175 . . . 4 ({𝑦} ∈ V → {𝑦} ∈ V)
107, 8, 93syl 17 . . 3 (𝑦𝐴 {𝑦} ∈ V)
1110adantl 262 . 2 ((Rel 𝐴𝑦𝐴) → {𝑦} ∈ V)
12 cnvf1olem 5845 . . 3 ((Rel 𝐴 ∧ (𝑥𝐴𝑦 = {𝑥})) → (𝑦𝐴𝑥 = {𝑦}))
13 relcnv 4703 . . . . 5 Rel 𝐴
14 simpr 103 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑦𝐴𝑥 = {𝑦}))
15 cnvf1olem 5845 . . . . 5 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
1613, 14, 15sylancr 393 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
17 dfrel2 4771 . . . . . . 7 (Rel 𝐴𝐴 = 𝐴)
18 eleq2 2101 . . . . . . 7 (𝐴 = 𝐴 → (𝑥𝐴𝑥𝐴))
1917, 18sylbi 114 . . . . . 6 (Rel 𝐴 → (𝑥𝐴𝑥𝐴))
2019anbi1d 438 . . . . 5 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2120adantr 261 . . . 4 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑥𝐴𝑦 = {𝑥})))
2216, 21mpbid 135 . . 3 ((Rel 𝐴 ∧ (𝑦𝐴𝑥 = {𝑦})) → (𝑥𝐴𝑦 = {𝑥}))
2312, 22impbida 528 . 2 (Rel 𝐴 → ((𝑥𝐴𝑦 = {𝑥}) ↔ (𝑦𝐴𝑥 = {𝑦})))
241, 6, 11, 23f1od 5703 1 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  ∪ cuni 3580   ↦ cmpt 3818  ◡ccnv 4344  Rel wrel 4350  –1-1-onto→wf1o 4901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1st 5767  df-2nd 5768 This theorem is referenced by:  tposf12  5884  cnven  6288  xpcomf1o  6299
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