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Theorem f1cnvcnv 5043
Description: Two ways to express that a set A (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1cnvcnv (A:dom A1-1→V ↔ (Fun A Fun A))

Proof of Theorem f1cnvcnv
StepHypRef Expression
1 df-f1 4850 . 2 (A:dom A1-1→V ↔ (A:dom A⟶V Fun A))
2 dffn2 4990 . . . 4 (A Fn dom AA:dom A⟶V)
3 dmcnvcnv 4501 . . . . 5 dom A = dom A
4 df-fn 4848 . . . . 5 (A Fn dom A ↔ (Fun A dom A = dom A))
53, 4mpbiran2 847 . . . 4 (A Fn dom A ↔ Fun A)
62, 5bitr3i 175 . . 3 (A:dom A⟶V ↔ Fun A)
7 relcnv 4646 . . . . 5 Rel A
8 dfrel2 4714 . . . . 5 (Rel AA = A)
97, 8mpbi 133 . . . 4 A = A
109funeqi 4865 . . 3 (Fun A ↔ Fun A)
116, 10anbi12ci 434 . 2 ((A:dom A⟶V Fun A) ↔ (Fun A Fun A))
121, 11bitri 173 1 (A:dom A1-1→V ↔ (Fun A Fun A))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  Vcvv 2551  ccnv 4287  dom cdm 4288  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840  wf 4841  1-1wf1 4842
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-bndl 1396  ax-4 1397  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
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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850
This theorem is referenced by: (None)
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