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Theorem oprabco 6884
 Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Hypotheses
Ref Expression
oprabco.1
oprabco.2
oprabco.3
Assertion
Ref Expression
oprabco
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem oprabco
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oprabco.1 . . . 4
21adantl 466 . . 3
3 oprabco.2 . . . 4
43a1i 11 . . 3
5 dffn5 5918 . . . 4
65biimpi 194 . . 3
7 fveq2 5871 . . 3
82, 4, 6, 7fmpt2co 6883 . 2
9 oprabco.3 . 2
108, 9syl6reqr 2517 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  e.cmpt 4510  o.ccom 5008  Fnwfn 5588  cfv 5593  e.`cmpt2 6298 This theorem is referenced by:  oprab2co  6885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801
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