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Theorem dmcoass 14223
 Description: The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o compa
coafval.a Nat
Assertion
Ref Expression
dmcoass

Proof of Theorem dmcoass
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coafval.o . . . 4 compa
2 coafval.a . . . 4 Nat
3 eqid 2438 . . . 4 comp comp
41, 2, 3coafval 14221 . . 3 coda coda compcoda
54dmmpt2ssx 6418 . 2 coda
6 iunss 4134 . . 3 coda coda
7 snssi 3944 . . . 4
8 ssrab2 3430 . . . 4 coda
9 xpss12 4983 . . . 4 coda coda
107, 8, 9sylancl 645 . . 3 coda
116, 10mprgbir 2778 . 2 coda
125, 11sstri 3359 1
 Colors of variables: wff set class Syntax hints:   wceq 1653   wcel 1726  crab 2711   wss 3322  csn 3816  cop 3819  cotp 3820  ciun 4095   cxp 4878   cdm 4880  cfv 5456  (class class class)co 6083  c2nd 6350  compcco 13543  cdoma 14177  codaccoda 14178  Natcarw 14179  compaccoa 14211 This theorem is referenced by:  coapm  14228 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-ot 3826  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-arw 14184  df-coa 14213
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