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Theorem caofdir 6344
 Description: Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
caofdi.1
caofdi.2
caofdi.3
caofdi.4
caofdir.5
Assertion
Ref Expression
caofdir
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem caofdir
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 caofdir.5 . . . . 5
21adantlr 697 . . . 4
3 caofdi.3 . . . . 5
43ffvelrnda 5873 . . . 4
5 caofdi.4 . . . . 5
65ffvelrnda 5873 . . . 4
7 caofdi.2 . . . . 5
87ffvelrnda 5873 . . . 4
92, 4, 6, 8caovdird 6268 . . 3
109mpteq2dva 4298 . 2
11 caofdi.1 . . 3
12 ovex 6109 . . . 4
1312a1i 11 . . 3
143feqmptd 5782 . . . 4
155feqmptd 5782 . . . 4
1611, 4, 6, 14, 15offval2 6325 . . 3
177feqmptd 5782 . . 3
1811, 13, 8, 16, 17offval2 6325 . 2
19 ovex 6109 . . . 4
2019a1i 11 . . 3
21 ovex 6109 . . . 4
2221a1i 11 . . 3
2311, 4, 8, 14, 17offval2 6325 . . 3
2411, 6, 8, 15, 17offval2 6325 . . 3
2511, 20, 22, 23, 24offval2 6325 . 2
2610, 18, 253eqtr4d 2480 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  cvv 2958   cmpt 4269  wf 5453  cfv 5457  (class class class)co 6084   cof 6306 This theorem is referenced by:  psrlmod  16470  mendlmod  27492  expgrowth  27543  lflvsdi1  29950 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308
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