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Theorem f1cnvcnv 5631
Description: Two ways to express that a set (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

Proof of Theorem f1cnvcnv
StepHypRef Expression
1 df-f1 5443 . 2
2 dffn2 5578 . . . 4
3 dmcnvcnv 5084 . . . . 5
4 df-fn 5441 . . . . 5
53, 4mpbiran2 887 . . . 4
62, 5bitr3i 244 . . 3
7 relcnv 5229 . . . . 5
8 dfrel2 5308 . . . . 5
97, 8mpbi 201 . . . 4
109funeqi 5458 . . 3
116, 10anbi12ci 681 . 2
121, 11bitri 242 1
Colors of variables: wff set class
Syntax hints:  <->wb 178  /\wa 360  =wceq 1670   cvv 3015  `'ccnv 4861  domcdm 4862  Relwrel 4867  Funwfun 5432  Fnwfn 5433  -->wf 5434  -1-1->wf1 5435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1570  ax-4 1581  ax-5 1644  ax-6 1685  ax-7 1705  ax-9 1736  ax-10 1751  ax-11 1756  ax-12 1768  ax-13 1955  ax-ext 2470  ax-sep 4439  ax-nul 4447  ax-pr 4554
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1338  df-ex 1566  df-nf 1569  df-sb 1677  df-eu 2317  df-mo 2318  df-clab 2476  df-cleq 2482  df-clel 2485  df-nfc 2614  df-ne 2654  df-ral 2764  df-rex 2765  df-rab 2768  df-v 3017  df-dif 3368  df-un 3370  df-in 3372  df-ss 3379  df-nul 3674  df-if 3826  df-sn 3915  df-pr 3916  df-op 3918  df-br 4319  df-opab 4377  df-xp 4868  df-rel 4869  df-cnv 4870  df-co 4871  df-dm 4872  df-rn 4873  df-fun 5440  df-fn 5441  df-f 5442  df-f1 5443
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