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Theorem comffval2 14633
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval2.o compf
comfffval2.b
comfffval2.h f
comfffval2.x comp
comffval2.x
comffval2.y
comffval2.z
Assertion
Ref Expression
comffval2
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem comffval2
StepHypRef Expression
1 comfffval2.o . . 3 compf
2 comfffval2.b . . 3
3 eqid 2438 . . 3
4 comfffval2.x . . 3 comp
5 comffval2.x . . 3
6 comffval2.y . . 3
7 comffval2.z . . 3
81, 2, 3, 4, 5, 6, 7comffval 14630 . 2
9 comfffval2.h . . . 4 f
109, 2, 3, 6, 7homfval 14623 . . 3
119, 2, 3, 5, 6homfval 14623 . . 3
12 eqidd 2439 . . 3
1310, 11, 12mpt2eq123dv 6143 . 2
148, 13eqtr4d 2473 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1369   wcel 1756  cop 3878  cfv 5413  (class class class)co 6086   cmpt2 6088  cbs 14166   chom 14241  compcco 14242   f chomf 14596  compfccomf 14597 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-homf 14600  df-comf 14601 This theorem is referenced by: (None)
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