Theorem f1ocnvb 3699

Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged.

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 3698	. 2 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
2		dfrel2 3482	. . . 4 |- (Rel F <-> `'`'F = F)
3		f1oeq1 3681	. . . 4 |- (`'`'F = F -> (`'`'F:A-1-1-onto->B <-> F:A-1-1-onto->B))
4	2, 3	sylbi 199	. . 3 |- (Rel F -> (`'`'F:A-1-1-onto->B <-> F:A-1-1-onto->B))
5		f1ocnv 3698	. . 3 |- (`'F:B-1-1-onto->A -> `'`'F:A-1-1-onto->B)
6	4, 5	syl5bi 208	. 2 |- (Rel F -> (`'F:B-1-1-onto->A -> F:A-1-1-onto->B))
7	1, 6	impbid2 517	1 |- (Rel F -> (F:A-1-1-onto->B <-> `'F:B-1-1-onto->A))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955  `'ccnv 3166  Rel wrel 3172  -1-1-onto->wf1o 3178

This theorem is referenced by:  cnvhmphb 10507  cnvhmph 10508

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194

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