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Theorem cnvf1o 5769
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 → (x A {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 2022 . 2 (x A {x}) = (x A {x})
2 snexg 3910 . . . 4 (x A → {x} V)
3 cnvexg 4782 . . . 4 ({x} V → {x} V)
4 uniexg 4125 . . . 4 ({x} V → {x} V)
52, 3, 43syl 17 . . 3 (x A {x} V)
65adantl 262 . 2 ((Rel A x A) → {x} V)
7 snexg 3910 . . . 4 (y A → {y} V)
8 cnvexg 4782 . . . 4 ({y} V → {y} V)
9 uniexg 4125 . . . 4 ({y} V → {y} V)
107, 8, 93syl 17 . . 3 (y A {y} V)
1110adantl 262 . 2 ((Rel A y A) → {y} V)
12 cnvf1olem 5768 . . 3 ((Rel A (x A y = {x})) → (y A x = {y}))
13 relcnv 4630 . . . . 5 Rel A
14 simpr 103 . . . . 5 ((Rel A (y A x = {y})) → (y A x = {y}))
15 cnvf1olem 5768 . . . . 5 ((Rel A (y A x = {y})) → (x A y = {x}))
1613, 14, 15sylancr 395 . . . 4 ((Rel A (y A x = {y})) → (x A y = {x}))
17 dfrel2 4698 . . . . . . 7 (Rel AA = A)
18 eleq2 2083 . . . . . . 7 (A = A → (x Ax A))
1917, 18sylbi 114 . . . . . 6 (Rel A → (x Ax A))
2019anbi1d 441 . . . . 5 (Rel A → ((x A y = {x}) ↔ (x A y = {x})))
2120adantr 261 . . . 4 ((Rel A (y A x = {y})) → ((x A y = {x}) ↔ (x A y = {x})))
2216, 21mpbid 135 . . 3 ((Rel A (y A x = {y})) → (x A y = {x}))
2312, 22impbida 515 . 2 (Rel A → ((x A y = {x}) ↔ (y A x = {y})))
241, 6, 11, 23f1od 5626 1 (Rel A → (x A {x}):A1-1-ontoA)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  {csn 3350   cuni 3554  cmpt 3792  ccnv 4271  Rel wrel 4277  1-1-ontowf1o 4828
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by:  tposf12  5806
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