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Theorem comfval 14635
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o compf
comfffval.b
comfffval.h
comfffval.x comp
comffval.x
comffval.y
comffval.z
comfval.f
comfval.g
Assertion
Ref Expression
comfval

Proof of Theorem comfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffval.o . . 3 compf
2 comfffval.b . . 3
3 comfffval.h . . 3
4 comfffval.x . . 3 comp
5 comffval.x . . 3
6 comffval.y . . 3
7 comffval.z . . 3
81, 2, 3, 4, 5, 6, 7comffval 14634 . 2
9 oveq12 6099 . . 3
109adantl 463 . 2
11 comfval.g . 2
12 comfval.f . 2
13 ovex 6115 . . 3
1413a1i 11 . 2
158, 10, 11, 12, 14ovmpt2d 6217 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1364   wcel 1761  cvv 2970  cop 3880  cfv 5415  (class class class)co 6090  cbs 14170   chom 14245  compcco 14246  compfccomf 14601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-comf 14605 This theorem is referenced by:  comfval2  14638  comfeqval  14643
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