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Theorem coapm 18020
Description: Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coapm.o Β· = (compaβ€˜πΆ)
coapm.a 𝐴 = (Arrowβ€˜πΆ)
Assertion
Ref Expression
coapm Β· ∈ (𝐴 ↑pm (𝐴 Γ— 𝐴))

Proof of Theorem coapm
Dummy variables 𝑓 𝑔 β„Ž 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coapm.o . . . . . 6 Β· = (compaβ€˜πΆ)
2 coapm.a . . . . . 6 𝐴 = (Arrowβ€˜πΆ)
3 eqid 2732 . . . . . 6 (compβ€˜πΆ) = (compβ€˜πΆ)
41, 2, 3coafval 18013 . . . . 5 Β· = (𝑔 ∈ 𝐴, 𝑓 ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} ↦ ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩(compβ€˜πΆ)(codaβ€˜π‘”))(2nd β€˜π‘“))⟩)
54mpofun 7531 . . . 4 Fun Β·
6 funfn 6578 . . . 4 (Fun Β· ↔ Β· Fn dom Β· )
75, 6mpbi 229 . . 3 Β· Fn dom Β·
81, 2dmcoass 18015 . . . . . . . . 9 dom Β· βŠ† (𝐴 Γ— 𝐴)
98sseli 3978 . . . . . . . 8 (𝑧 ∈ dom Β· β†’ 𝑧 ∈ (𝐴 Γ— 𝐴))
10 1st2nd2 8013 . . . . . . . 8 (𝑧 ∈ (𝐴 Γ— 𝐴) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
119, 10syl 17 . . . . . . 7 (𝑧 ∈ dom Β· β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
1211fveq2d 6895 . . . . . 6 (𝑧 ∈ dom Β· β†’ ( Β· β€˜π‘§) = ( Β· β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
13 df-ov 7411 . . . . . 6 ((1st β€˜π‘§) Β· (2nd β€˜π‘§)) = ( Β· β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
1412, 13eqtr4di 2790 . . . . 5 (𝑧 ∈ dom Β· β†’ ( Β· β€˜π‘§) = ((1st β€˜π‘§) Β· (2nd β€˜π‘§)))
15 eqid 2732 . . . . . . 7 (Homaβ€˜πΆ) = (Homaβ€˜πΆ)
162, 15homarw 17995 . . . . . 6 ((domaβ€˜(2nd β€˜π‘§))(Homaβ€˜πΆ)(codaβ€˜(1st β€˜π‘§))) βŠ† 𝐴
17 id 22 . . . . . . . . . . . . 13 (𝑧 ∈ dom Β· β†’ 𝑧 ∈ dom Β· )
1811, 17eqeltrrd 2834 . . . . . . . . . . . 12 (𝑧 ∈ dom Β· β†’ ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∈ dom Β· )
19 df-br 5149 . . . . . . . . . . . 12 ((1st β€˜π‘§)dom Β· (2nd β€˜π‘§) ↔ ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ ∈ dom Β· )
2018, 19sylibr 233 . . . . . . . . . . 11 (𝑧 ∈ dom Β· β†’ (1st β€˜π‘§)dom Β· (2nd β€˜π‘§))
211, 2eldmcoa 18014 . . . . . . . . . . 11 ((1st β€˜π‘§)dom Β· (2nd β€˜π‘§) ↔ ((2nd β€˜π‘§) ∈ 𝐴 ∧ (1st β€˜π‘§) ∈ 𝐴 ∧ (codaβ€˜(2nd β€˜π‘§)) = (domaβ€˜(1st β€˜π‘§))))
2220, 21sylib 217 . . . . . . . . . 10 (𝑧 ∈ dom Β· β†’ ((2nd β€˜π‘§) ∈ 𝐴 ∧ (1st β€˜π‘§) ∈ 𝐴 ∧ (codaβ€˜(2nd β€˜π‘§)) = (domaβ€˜(1st β€˜π‘§))))
2322simp1d 1142 . . . . . . . . 9 (𝑧 ∈ dom Β· β†’ (2nd β€˜π‘§) ∈ 𝐴)
242, 15arwhoma 17994 . . . . . . . . 9 ((2nd β€˜π‘§) ∈ 𝐴 β†’ (2nd β€˜π‘§) ∈ ((domaβ€˜(2nd β€˜π‘§))(Homaβ€˜πΆ)(codaβ€˜(2nd β€˜π‘§))))
2523, 24syl 17 . . . . . . . 8 (𝑧 ∈ dom Β· β†’ (2nd β€˜π‘§) ∈ ((domaβ€˜(2nd β€˜π‘§))(Homaβ€˜πΆ)(codaβ€˜(2nd β€˜π‘§))))
2622simp3d 1144 . . . . . . . . 9 (𝑧 ∈ dom Β· β†’ (codaβ€˜(2nd β€˜π‘§)) = (domaβ€˜(1st β€˜π‘§)))
2726oveq2d 7424 . . . . . . . 8 (𝑧 ∈ dom Β· β†’ ((domaβ€˜(2nd β€˜π‘§))(Homaβ€˜πΆ)(codaβ€˜(2nd β€˜π‘§))) = ((domaβ€˜(2nd β€˜π‘§))(Homaβ€˜πΆ)(domaβ€˜(1st β€˜π‘§))))
2825, 27eleqtrd 2835 . . . . . . 7 (𝑧 ∈ dom Β· β†’ (2nd β€˜π‘§) ∈ ((domaβ€˜(2nd β€˜π‘§))(Homaβ€˜πΆ)(domaβ€˜(1st β€˜π‘§))))
2922simp2d 1143 . . . . . . . 8 (𝑧 ∈ dom Β· β†’ (1st β€˜π‘§) ∈ 𝐴)
302, 15arwhoma 17994 . . . . . . . 8 ((1st β€˜π‘§) ∈ 𝐴 β†’ (1st β€˜π‘§) ∈ ((domaβ€˜(1st β€˜π‘§))(Homaβ€˜πΆ)(codaβ€˜(1st β€˜π‘§))))
3129, 30syl 17 . . . . . . 7 (𝑧 ∈ dom Β· β†’ (1st β€˜π‘§) ∈ ((domaβ€˜(1st β€˜π‘§))(Homaβ€˜πΆ)(codaβ€˜(1st β€˜π‘§))))
321, 15, 28, 31coahom 18019 . . . . . 6 (𝑧 ∈ dom Β· β†’ ((1st β€˜π‘§) Β· (2nd β€˜π‘§)) ∈ ((domaβ€˜(2nd β€˜π‘§))(Homaβ€˜πΆ)(codaβ€˜(1st β€˜π‘§))))
3316, 32sselid 3980 . . . . 5 (𝑧 ∈ dom Β· β†’ ((1st β€˜π‘§) Β· (2nd β€˜π‘§)) ∈ 𝐴)
3414, 33eqeltrd 2833 . . . 4 (𝑧 ∈ dom Β· β†’ ( Β· β€˜π‘§) ∈ 𝐴)
3534rgen 3063 . . 3 βˆ€π‘§ ∈ dom Β· ( Β· β€˜π‘§) ∈ 𝐴
36 ffnfv 7117 . . 3 ( Β· :dom Β· ⟢𝐴 ↔ ( Β· Fn dom Β· ∧ βˆ€π‘§ ∈ dom Β· ( Β· β€˜π‘§) ∈ 𝐴))
377, 35, 36mpbir2an 709 . 2 Β· :dom Β· ⟢𝐴
382fvexi 6905 . . 3 𝐴 ∈ V
3938, 38xpex 7739 . . 3 (𝐴 Γ— 𝐴) ∈ V
4038, 39elpm2 8867 . 2 ( Β· ∈ (𝐴 ↑pm (𝐴 Γ— 𝐴)) ↔ ( Β· :dom Β· ⟢𝐴 ∧ dom Β· βŠ† (𝐴 Γ— 𝐴)))
4137, 8, 40mpbir2an 709 1 Β· ∈ (𝐴 ↑pm (𝐴 Γ— 𝐴))
Colors of variables: wff setvar class
Syntax hints:   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βŠ† wss 3948  βŸ¨cop 4634  βŸ¨cotp 4636   class class class wbr 5148   Γ— cxp 5674  dom cdm 5676  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973   ↑pm cpm 8820  compcco 17208  domacdoma 17969  codaccoda 17970  Arrowcarw 17971  Homachoma 17972  compaccoa 18003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-pm 8822  df-cat 17611  df-doma 17973  df-coda 17974  df-homa 17975  df-arw 17976  df-coa 18005
This theorem is referenced by: (None)
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