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Theorem coafval 14219
 Description: The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o compa
coafval.a Nat
coafval.x comp
Assertion
Ref Expression
coafval coda coda coda
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem coafval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 coafval.o . 2 compa
2 fveq2 5728 . . . . . 6 Nat Nat
3 coafval.a . . . . . 6 Nat
42, 3syl6eqr 2486 . . . . 5 Nat
5 biidd 229 . . . . . 6 coda coda
64, 5rabeqbidv 2951 . . . . 5 Nat coda coda
7 fveq2 5728 . . . . . . . . 9 comp comp
8 coafval.x . . . . . . . . 9 comp
97, 8syl6eqr 2486 . . . . . . . 8 comp
109oveqd 6098 . . . . . . 7 compcoda coda
1110oveqd 6098 . . . . . 6 compcoda coda
1211oteq3d 3998 . . . . 5 coda compcoda coda coda
134, 6, 12mpt2eq123dv 6136 . . . 4 Nat Nat coda coda compcoda coda coda coda
14 df-coa 14211 . . . 4 compa Nat Nat coda coda compcoda
15 fvex 5742 . . . . . 6 Nat
163, 15eqeltri 2506 . . . . 5
1716rabex 4354 . . . . 5 coda
1816, 17mpt2ex 6425 . . . 4 coda coda coda
1913, 14, 18fvmpt 5806 . . 3 compa coda coda coda
2014dmmptss 5366 . . . . . . 7 compa
2120sseli 3344 . . . . . 6 compa
2221con3i 129 . . . . 5 compa
23 ndmfv 5755 . . . . 5 compa compa
2422, 23syl 16 . . . 4 compa
253arwrcl 14199 . . . . . . . 8
2625con3i 129 . . . . . . 7
2726eq0rdv 3662 . . . . . 6
28 eqidd 2437 . . . . . 6 coda coda
29 eqidd 2437 . . . . . 6 coda coda coda coda
3027, 28, 29mpt2eq123dv 6136 . . . . 5 coda coda coda coda coda coda
31 mpt20 6427 . . . . 5 coda coda coda
3230, 31syl6eq 2484 . . . 4 coda coda coda
3324, 32eqtr4d 2471 . . 3 compa coda coda coda
3419, 33pm2.61i 158 . 2 compa coda coda coda
351, 34eqtri 2456 1 coda coda coda
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1652   wcel 1725  crab 2709  cvv 2956  c0 3628  cop 3817  cotp 3818   cdm 4878  cfv 5454  (class class class)co 6081   cmpt2 6083  c2nd 6348  compcco 13541  ccat 13889  cdoma 14175  codaccoda 14176  Natcarw 14177  compaccoa 14209 This theorem is referenced by:  eldmcoa  14220  dmcoass  14221  coaval  14223  coapm  14226 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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-arw 14182  df-coa 14211
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