Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  comfeqval Structured version   Unicode version

Theorem comfeqval 14981
 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 6312 . . 3 compf compf
32oveqd 6312 . 2 compf compf
4 eqid 2467 . . 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 14973 . 2 compf
14 eqid 2467 . . 3 compf compf
15 eqid 2467 . . 3
16 eqid 2467 . . 3
17 comfeqval.2 . . 3 comp
18 comfeqval.3 . . . . . 6 f f
1918homfeqbas 14969 . . . . 5
205, 19syl5eq 2520 . . . 4
218, 20eleqtrd 2557 . . 3
229, 20eleqtrd 2557 . . 3
2310, 20eleqtrd 2557 . . 3
245, 6, 16, 18, 8, 9homfeqval 14970 . . . 4
2511, 24eleqtrd 2557 . . 3
265, 6, 16, 18, 9, 10homfeqval 14970 . . . 4
2712, 26eleqtrd 2557 . . 3
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 14973 . 2 compf
293, 13, 283eqtr3d 2516 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1379   wcel 1767  cop 4039  cfv 5594  (class class class)co 6295  cbs 14507   chom 14583  compcco 14584   f chomf 14938  compfccomf 14939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-homf 14942  df-comf 14943 This theorem is referenced by:  catpropd  14982  cidpropd  14983  oppccomfpropd  15000  monpropd  15010  funcpropd  15144  natpropd  15220  fucpropd  15221  xpcpropd  15352  hofpropd  15411
 Copyright terms: Public domain W3C validator