Theorem cocnvcnv2 3503

Description: A composition is not affected by a double converse of its second argument.

Assertion

Ref	Expression
cocnvcnv2	|- (A o. `'`'B) = (A o. B)

Proof of Theorem cocnvcnv2

Step	Hyp	Ref	Expression
1		cnvcnv2 3484	. . 3 |- `'`'B = (B |` V)
2	1	coeq2i 3281	. 2 |- (A o. `'`'B) = (A o. (B |` V))
3		resco 3497	. 2 |- ((A o. B) |` V) = (A o. (B |` V))
4		relco 3481	. . 3 |- Rel (A o. B)
5		dfrel3 3486	. . 3 |- (Rel (A o. B) <-> ((A o. B) |` V) = (A o. B))
6	4, 5	mpbi 189	. 2 |- ((A o. B) |` V) = (A o. B)
7	2, 3, 6	3eqtr2 1500	1 |- (A o. `'`'B) = (A o. B)

Syntax hints:   = wceq 955  Vcvv 1809  `'ccnv 3166   |` cres 3169   o. ccom 3171  Rel wrel 3172

This theorem is referenced by:  cofunex2g 3578

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-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-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-res 3187

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