Theorem f1ocnvb 5597

Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/range interchanged. (Contributed by NM, 8-Dec-2003.)

Assertion

Ref	Expression
f1ocnvb	|- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) )

Proof of Theorem f1ocnvb

Step	Hyp	Ref	Expression
1		f1ocnv 5596	. 2 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1ocnv 5596	. . 3 |- ( `' F : B -1-1-onto-> A -> `' `' F : A -1-1-onto-> B )
3		dfrel2 5187	. . . 4 |- ( Rel F <-> `' `' F = F )
4		f1oeq1 5571	. . . 4 |- ( `' `' F = F -> ( `' `' F : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
5	3, 4	sylbi 121	. . 3 |- ( Rel F -> ( `' `' F : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
6	2, 5	imbitrid 154	. 2 |- ( Rel F -> ( `' F : B -1-1-onto-> A -> F : A -1-1-onto-> B ) )
7	1, 6	impbid2 143	1 |- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   `'ccnv 4724   Rel wrel 4730   -1-1-onto->wf1o 5325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by: (None)