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Theorem cnvf1o 5788
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 2037 . 2 (x A {x}) = (x A {x})
2 snexg 3927 . . . 4 (x A → {x} V)
3 cnvexg 4798 . . . 4 ({x} V → {x} V)
4 uniexg 4141 . . . 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 3927 . . . 4 (y A → {y} V)
8 cnvexg 4798 . . . 4 ({y} V → {y} V)
9 uniexg 4141 . . . 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 5787 . . 3 ((Rel A (x A y = {x})) → (y A x = {y}))
13 relcnv 4646 . . . . 5 Rel A
14 simpr 103 . . . . 5 ((Rel A (y A x = {y})) → (y A x = {y}))
15 cnvf1olem 5787 . . . . 5 ((Rel A (y A x = {y})) → (x A y = {x}))
1613, 14, 15sylancr 393 . . . 4 ((Rel A (y A x = {y})) → (x A y = {x}))
17 dfrel2 4714 . . . . . . 7 (Rel AA = A)
18 eleq2 2098 . . . . . . 7 (A = A → (x Ax A))
1917, 18sylbi 114 . . . . . 6 (Rel A → (x Ax A))
2019anbi1d 438 . . . . 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 528 . 2 (Rel A → ((x A y = {x}) ↔ (y A x = {y})))
241, 6, 11, 23f1od 5645 1 (Rel A → (x A {x}):A1-1-ontoA)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367   cuni 3571  cmpt 3809  ccnv 4287  Rel wrel 4293  1-1-ontowf1o 4844
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  tposf12  5825  cnven  6224  xpcomf1o  6235
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