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Theorem comfeqval 13926
 Description: Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfeqval.b
comfeqval.h
comfeqval.1 comp
comfeqval.2 comp
comfeqval.3 f f
comfeqval.4 compf compf
comfeqval.x
comfeqval.y
comfeqval.z
comfeqval.f
comfeqval.g
Assertion
Ref Expression
comfeqval

Proof of Theorem comfeqval
StepHypRef Expression
1 comfeqval.4 . . . 4 compf compf
21oveqd 6090 . . 3 compf compf
32oveqd 6090 . 2 compf compf
4 eqid 2435 . . 3 compf compf
5 comfeqval.b . . 3
6 comfeqval.h . . 3
7 comfeqval.1 . . 3 comp
8 comfeqval.x . . 3
9 comfeqval.y . . 3
10 comfeqval.z . . 3
11 comfeqval.f . . 3
12 comfeqval.g . . 3
134, 5, 6, 7, 8, 9, 10, 11, 12comfval 13918 . 2 compf
14 eqid 2435 . . 3 compf compf
15 eqid 2435 . . 3
16 eqid 2435 . . 3
17 comfeqval.2 . . 3 comp
18 comfeqval.3 . . . . . 6 f f
1918homfeqbas 13914 . . . . 5
205, 19syl5eq 2479 . . . 4
218, 20eleqtrd 2511 . . 3
229, 20eleqtrd 2511 . . 3
2310, 20eleqtrd 2511 . . 3
245, 6, 16, 18, 8, 9homfeqval 13915 . . . 4
2511, 24eleqtrd 2511 . . 3
265, 6, 16, 18, 9, 10homfeqval 13915 . . . 4
2712, 26eleqtrd 2511 . . 3
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 13918 . 2 compf
293, 13, 283eqtr3d 2475 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cop 3809  cfv 5446  (class class class)co 6073  cbs 13461   chom 13532  compcco 13533   f chomf 13883  compfccomf 13884 This theorem is referenced by:  catpropd  13927  cidpropd  13928  oppccomfpropd  13945  monpropd  13955  funcpropd  14089  natpropd  14165  fucpropd  14166  xpcpropd  14297  hofpropd  14356 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-homf 13887  df-comf 13888
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