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Theorem catcco 18065
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 18064 . . 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 768 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑣 = βŸ¨π‘‹, π‘ŒβŸ©)
76fveq2d 6888 . . . . . 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 7984 . . . . . . . 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 2766 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = π‘Œ)
14 simprr 770 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑧 = 𝑍)
1513, 14oveq12d 7422 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((2nd β€˜π‘£) Func 𝑧) = (π‘Œ Func 𝑍))
166fveq2d 6888 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ( Func β€˜π‘£) = ( Func β€˜βŸ¨π‘‹, π‘ŒβŸ©))
17 df-ov 7407 . . . . 5 (𝑋 Func π‘Œ) = ( Func β€˜βŸ¨π‘‹, π‘ŒβŸ©)
1816, 17eqtr4di 2784 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ( Func β€˜π‘£) = (𝑋 Func π‘Œ))
19 eqidd 2727 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∘func 𝑓) = (𝑔 ∘func 𝑓))
2015, 18, 19mpoeq123dv 7479 . . 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 5708 . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝐡 Γ— 𝐡))
22 catcco.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
23 ovex 7437 . . . . 5 (π‘Œ Func 𝑍) ∈ V
24 ovex 7437 . . . . 5 (𝑋 Func π‘Œ) ∈ V
2523, 24mpoex 8062 . . . 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 7555 . 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 7413 . . 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 7439 . 2 (πœ‘ β†’ (𝐺 ∘func 𝐹) ∈ V)
3327, 29, 30, 31, 32ovmpod 7555 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 1533   ∈ wcel 2098  Vcvv 3468  βŸ¨cop 4629   Γ— cxp 5667  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  2nd c2nd 7970  Basecbs 17151  compcco 17216   Func cfunc 17811   ∘func ccofu 17813  CatCatccatc 18058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-hom 17228  df-cco 17229  df-catc 18059
This theorem is referenced by:  catccatid  18066  resscatc  18069  catcisolem  18070  catciso  18071
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