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Theorem fmptcos 5634
 Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fmptcof.1
fmptcof.2
fmptcof.3
Assertion
Ref Expression
fmptcos
Distinct variable groups:   ,,   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()   ()   (,)   (,)

Proof of Theorem fmptcos
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fmptcof.1 . 2
2 fmptcof.2 . 2
3 fmptcof.3 . . 3
4 nfcv 2299 . . . 4
5 nfcsb1v 3064 . . . 4
6 csbeq1a 3040 . . . 4
74, 5, 6cbvmpt 4059 . . 3
83, 7eqtrdi 2206 . 2
9 csbeq1 3034 . 2
101, 2, 8, 9fmptcof 5633 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1335   wcel 2128  wral 2435  csb 3031   cmpt 4025   ccom 4589 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177 This theorem is referenced by:  fmpoco  6160
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