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Theorem catcco 18094
Description: Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catcbas.c 𝐢 = (CatCatβ€˜π‘ˆ)
catcbas.b 𝐡 = (Baseβ€˜πΆ)
catcbas.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
catcco.o Β· = (compβ€˜πΆ)
catcco.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
catcco.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
catcco.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
catcco.f (πœ‘ β†’ 𝐹 ∈ (𝑋 Func π‘Œ))
catcco.g (πœ‘ β†’ 𝐺 ∈ (π‘Œ Func 𝑍))
Assertion
Ref Expression
catcco (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘func 𝐹))

Proof of Theorem catcco
Dummy variables 𝑣 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcbas.c . . . 4 𝐢 = (CatCatβ€˜π‘ˆ)
2 catcbas.b . . . 4 𝐡 = (Baseβ€˜πΆ)
3 catcbas.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
4 catcco.o . . . 4 Β· = (compβ€˜πΆ)
51, 2, 3, 4catccofval 18093 . . 3 (πœ‘ β†’ Β· = (𝑣 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓))))
6 simprl 770 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑣 = βŸ¨π‘‹, π‘ŒβŸ©)
76fveq2d 6901 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©))
8 catcco.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝐡)
9 catcco.y . . . . . . . 8 (πœ‘ β†’ π‘Œ ∈ 𝐡)
10 op2ndg 8006 . . . . . . . 8 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
118, 9, 10syl2anc 583 . . . . . . 7 (πœ‘ β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
1211adantr 480 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
137, 12eqtrd 2768 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = π‘Œ)
14 simprr 772 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑧 = 𝑍)
1513, 14oveq12d 7438 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((2nd β€˜π‘£) Func 𝑧) = (π‘Œ Func 𝑍))
166fveq2d 6901 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ( Func β€˜π‘£) = ( Func β€˜βŸ¨π‘‹, π‘ŒβŸ©))
17 df-ov 7423 . . . . 5 (𝑋 Func π‘Œ) = ( Func β€˜βŸ¨π‘‹, π‘ŒβŸ©)
1816, 17eqtr4di 2786 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ( Func β€˜π‘£) = (𝑋 Func π‘Œ))
19 eqidd 2729 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∘func 𝑓) = (𝑔 ∘func 𝑓))
2015, 18, 19mpoeq123dv 7495 . . 3 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)) = (𝑔 ∈ (π‘Œ Func 𝑍), 𝑓 ∈ (𝑋 Func π‘Œ) ↦ (𝑔 ∘func 𝑓)))
218, 9opelxpd 5717 . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝐡 Γ— 𝐡))
22 catcco.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
23 ovex 7453 . . . . 5 (π‘Œ Func 𝑍) ∈ V
24 ovex 7453 . . . . 5 (𝑋 Func π‘Œ) ∈ V
2523, 24mpoex 8084 . . . 4 (𝑔 ∈ (π‘Œ Func 𝑍), 𝑓 ∈ (𝑋 Func π‘Œ) ↦ (𝑔 ∘func 𝑓)) ∈ V
2625a1i 11 . . 3 (πœ‘ β†’ (𝑔 ∈ (π‘Œ Func 𝑍), 𝑓 ∈ (𝑋 Func π‘Œ) ↦ (𝑔 ∘func 𝑓)) ∈ V)
275, 20, 21, 22, 26ovmpod 7573 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍) = (𝑔 ∈ (π‘Œ Func 𝑍), 𝑓 ∈ (𝑋 Func π‘Œ) ↦ (𝑔 ∘func 𝑓)))
28 oveq12 7429 . . 3 ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) β†’ (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹))
2928adantl 481 . 2 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹))
30 catcco.g . 2 (πœ‘ β†’ 𝐺 ∈ (π‘Œ Func 𝑍))
31 catcco.f . 2 (πœ‘ β†’ 𝐹 ∈ (𝑋 Func π‘Œ))
32 ovexd 7455 . 2 (πœ‘ β†’ (𝐺 ∘func 𝐹) ∈ V)
3327, 29, 30, 31, 32ovmpod 7573 1 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘func 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471  βŸ¨cop 4635   Γ— cxp 5676  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422  2nd c2nd 7992  Basecbs 17180  compcco 17245   Func cfunc 17840   ∘func ccofu 17842  CatCatccatc 18087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-fz 13518  df-struct 17116  df-slot 17151  df-ndx 17163  df-base 17181  df-hom 17257  df-cco 17258  df-catc 18088
This theorem is referenced by:  catccatid  18095  resscatc  18098  catcisolem  18099  catciso  18100
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