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Theorem subccocl 17791
Description: A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcidcl.j (πœ‘ β†’ 𝐽 ∈ (Subcatβ€˜πΆ))
subcidcl.2 (πœ‘ β†’ 𝐽 Fn (𝑆 Γ— 𝑆))
subcidcl.x (πœ‘ β†’ 𝑋 ∈ 𝑆)
subccocl.o Β· = (compβ€˜πΆ)
subccocl.y (πœ‘ β†’ π‘Œ ∈ 𝑆)
subccocl.z (πœ‘ β†’ 𝑍 ∈ 𝑆)
subccocl.f (πœ‘ β†’ 𝐹 ∈ (π‘‹π½π‘Œ))
subccocl.g (πœ‘ β†’ 𝐺 ∈ (π‘Œπ½π‘))
Assertion
Ref Expression
subccocl (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍))

Proof of Theorem subccocl
Dummy variables 𝑓 𝑔 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcidcl.j . . . 4 (πœ‘ β†’ 𝐽 ∈ (Subcatβ€˜πΆ))
2 eqid 2732 . . . . 5 (Homf β€˜πΆ) = (Homf β€˜πΆ)
3 eqid 2732 . . . . 5 (Idβ€˜πΆ) = (Idβ€˜πΆ)
4 subccocl.o . . . . 5 Β· = (compβ€˜πΆ)
5 subcrcl 17759 . . . . . 6 (𝐽 ∈ (Subcatβ€˜πΆ) β†’ 𝐢 ∈ Cat)
61, 5syl 17 . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
7 subcidcl.2 . . . . 5 (πœ‘ β†’ 𝐽 Fn (𝑆 Γ— 𝑆))
82, 3, 4, 6, 7issubc2 17782 . . . 4 (πœ‘ β†’ (𝐽 ∈ (Subcatβ€˜πΆ) ↔ (𝐽 βŠ†cat (Homf β€˜πΆ) ∧ βˆ€π‘₯ ∈ 𝑆 (((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝑆 βˆ€π‘§ ∈ 𝑆 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧)))))
91, 8mpbid 231 . . 3 (πœ‘ β†’ (𝐽 βŠ†cat (Homf β€˜πΆ) ∧ βˆ€π‘₯ ∈ 𝑆 (((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝑆 βˆ€π‘§ ∈ 𝑆 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧))))
109simprd 496 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑆 (((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝑆 βˆ€π‘§ ∈ 𝑆 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧)))
11 subcidcl.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑆)
12 subccocl.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑆)
1312adantr 481 . . . . 5 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ π‘Œ ∈ 𝑆)
14 subccocl.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ 𝑆)
1514ad2antrr 724 . . . . . 6 (((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) β†’ 𝑍 ∈ 𝑆)
16 subccocl.f . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ (π‘‹π½π‘Œ))
1716ad3antrrr 728 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) β†’ 𝐹 ∈ (π‘‹π½π‘Œ))
18 simpllr 774 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) β†’ π‘₯ = 𝑋)
19 simplr 767 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) β†’ 𝑦 = π‘Œ)
2018, 19oveq12d 7423 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) β†’ (π‘₯𝐽𝑦) = (π‘‹π½π‘Œ))
2117, 20eleqtrrd 2836 . . . . . . 7 ((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) β†’ 𝐹 ∈ (π‘₯𝐽𝑦))
22 subccocl.g . . . . . . . . . 10 (πœ‘ β†’ 𝐺 ∈ (π‘Œπ½π‘))
2322ad4antr 730 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) β†’ 𝐺 ∈ (π‘Œπ½π‘))
24 simpllr 774 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) β†’ 𝑦 = π‘Œ)
25 simplr 767 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) β†’ 𝑧 = 𝑍)
2624, 25oveq12d 7423 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) β†’ (𝑦𝐽𝑧) = (π‘Œπ½π‘))
2723, 26eleqtrrd 2836 . . . . . . . 8 (((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) β†’ 𝐺 ∈ (𝑦𝐽𝑧))
28 simp-5r 784 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ π‘₯ = 𝑋)
29 simp-4r 782 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ 𝑦 = π‘Œ)
3028, 29opeq12d 4880 . . . . . . . . . . 11 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ ⟨π‘₯, π‘¦βŸ© = βŸ¨π‘‹, π‘ŒβŸ©)
31 simpllr 774 . . . . . . . . . . 11 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ 𝑧 = 𝑍)
3230, 31oveq12d 7423 . . . . . . . . . 10 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ (⟨π‘₯, π‘¦βŸ© Β· 𝑧) = (βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍))
33 simpr 485 . . . . . . . . . 10 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
34 simplr 767 . . . . . . . . . 10 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ 𝑓 = 𝐹)
3532, 33, 34oveq123d 7426 . . . . . . . . 9 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ (𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) = (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹))
3628, 31oveq12d 7423 . . . . . . . . 9 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ (π‘₯𝐽𝑧) = (𝑋𝐽𝑍))
3735, 36eleq12d 2827 . . . . . . . 8 ((((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) β†’ ((𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧) ↔ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
3827, 37rspcdv 3604 . . . . . . 7 (((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) β†’ (βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧) β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
3921, 38rspcimdv 3602 . . . . . 6 ((((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) ∧ 𝑧 = 𝑍) β†’ (βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧) β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4015, 39rspcimdv 3602 . . . . 5 (((πœ‘ ∧ π‘₯ = 𝑋) ∧ 𝑦 = π‘Œ) β†’ (βˆ€π‘§ ∈ 𝑆 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧) β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4113, 40rspcimdv 3602 . . . 4 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ (βˆ€π‘¦ ∈ 𝑆 βˆ€π‘§ ∈ 𝑆 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧) β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4241adantld 491 . . 3 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ ((((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝑆 βˆ€π‘§ ∈ 𝑆 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧)) β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4311, 42rspcimdv 3602 . 2 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑆 (((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ βˆ€π‘¦ ∈ 𝑆 βˆ€π‘§ ∈ 𝑆 βˆ€π‘“ ∈ (π‘₯𝐽𝑦)βˆ€π‘” ∈ (𝑦𝐽𝑧)(𝑔(⟨π‘₯, π‘¦βŸ© Β· 𝑧)𝑓) ∈ (π‘₯𝐽𝑧)) β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4410, 43mpd 15 1 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) ∈ (𝑋𝐽𝑍))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βŸ¨cop 4633   class class class wbr 5147   Γ— cxp 5673   Fn wfn 6535  β€˜cfv 6540  (class class class)co 7405  compcco 17205  Catccat 17604  Idccid 17605  Homf chomf 17606   βŠ†cat cssc 17750  Subcatcsubc 17752
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-pm 8819  df-ixp 8888  df-ssc 17753  df-subc 17755
This theorem is referenced by:  subccatid  17792  funcres  17842
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