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Theorem caofcom 6109
 Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
caofref.1
caofref.2
caofcom.3
caofcom.4
Assertion
Ref Expression
caofcom
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem caofcom
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . . 6
2 ffvelrn 5663 . . . . . 6
31, 2sylan 457 . . . . 5
4 caofcom.3 . . . . . 6
5 ffvelrn 5663 . . . . . 6
64, 5sylan 457 . . . . 5
73, 6jca 518 . . . 4
8 caofcom.4 . . . . 5
98caovcomg 6015 . . . 4
107, 9syldan 456 . . 3
1110mpteq2dva 4106 . 2
12 caofref.1 . . 3
131feqmptd 5575 . . 3
144feqmptd 5575 . . 3
1512, 3, 6, 13, 14offval2 6095 . 2
1612, 6, 3, 14, 13offval2 6095 . 2
1711, 15, 163eqtr4d 2325 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684   cmpt 4077  wf 5251  cfv 5255  (class class class)co 5858   cof 6076 This theorem is referenced by:  plydivlem4  19676  quotcan  19689  dchrabl  20493  expgrowth  27552  lfladdcom  29262 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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