Theorem cocnvcnv2 3506

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 3487	. . 3 |- `'`'B = (B |` V)
2	1	coeq2i 3284	. 2 |- (A o. `'`'B) = (A o. (B |` V))
3		resco 3500	. 2 |- ((A o. B) |` V) = (A o. (B |` V))
4		relco 3484	. . 3 |- Rel (A o. B)
5		dfrel3 3489	. . 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 1501	1 |- (A o. `'`'B) = (A o. B)

Syntax hints:   = wceq 956  Vcvv 1811  `'ccnv 3169   |` cres 3172   o. ccom 3174  Rel wrel 3175

This theorem is referenced by:  cofunex2g 3581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-res 3190

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