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Theorem setcco 17929
Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcbas.c 𝐢 = (SetCatβ€˜π‘ˆ)
setcbas.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
setcco.o Β· = (compβ€˜πΆ)
setcco.x (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
setcco.y (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
setcco.z (πœ‘ β†’ 𝑍 ∈ π‘ˆ)
setcco.f (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
setcco.g (πœ‘ β†’ 𝐺:π‘ŒβŸΆπ‘)
Assertion
Ref Expression
setcco (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Proof of Theorem setcco
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 setcbas.c . . . 4 𝐢 = (SetCatβ€˜π‘ˆ)
2 setcbas.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
3 setcco.o . . . 4 Β· = (compβ€˜πΆ)
41, 2, 3setccofval 17928 . . 3 (πœ‘ β†’ Β· = (𝑣 ∈ (π‘ˆ Γ— π‘ˆ), 𝑧 ∈ π‘ˆ ↦ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))))
5 simprr 771 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑧 = 𝑍)
6 simprl 769 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑣 = βŸ¨π‘‹, π‘ŒβŸ©)
76fveq2d 6843 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©))
8 setcco.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
9 setcco.y . . . . . . . 8 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
10 op2ndg 7926 . . . . . . . 8 ((𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
118, 9, 10syl2anc 584 . . . . . . 7 (πœ‘ β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
1211adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
137, 12eqtrd 2777 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = π‘Œ)
145, 13oveq12d 7369 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑧 ↑m (2nd β€˜π‘£)) = (𝑍 ↑m π‘Œ))
156fveq2d 6843 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©))
16 op1stg 7925 . . . . . . . 8 ((𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
178, 9, 16syl2anc 584 . . . . . . 7 (πœ‘ β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
1817adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
1915, 18eqtrd 2777 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = 𝑋)
2013, 19oveq12d 7369 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) = (π‘Œ ↑m 𝑋))
21 eqidd 2738 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
2214, 20, 21mpoeq123dv 7426 . . 3 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑍 ↑m π‘Œ), 𝑓 ∈ (π‘Œ ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)))
238, 9opelxpd 5669 . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (π‘ˆ Γ— π‘ˆ))
24 setcco.z . . 3 (πœ‘ β†’ 𝑍 ∈ π‘ˆ)
25 ovex 7384 . . . . 5 (𝑍 ↑m π‘Œ) ∈ V
26 ovex 7384 . . . . 5 (π‘Œ ↑m 𝑋) ∈ V
2725, 26mpoex 8004 . . . 4 (𝑔 ∈ (𝑍 ↑m π‘Œ), 𝑓 ∈ (π‘Œ ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V
2827a1i 11 . . 3 (πœ‘ β†’ (𝑔 ∈ (𝑍 ↑m π‘Œ), 𝑓 ∈ (π‘Œ ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V)
294, 22, 23, 24, 28ovmpod 7501 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍) = (𝑔 ∈ (𝑍 ↑m π‘Œ), 𝑓 ∈ (π‘Œ ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)))
30 simprl 769 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑔 = 𝐺)
31 simprr 771 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑓 = 𝐹)
3230, 31coeq12d 5818 . 2 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
33 setcco.g . . 3 (πœ‘ β†’ 𝐺:π‘ŒβŸΆπ‘)
3424, 9elmapd 8737 . . 3 (πœ‘ β†’ (𝐺 ∈ (𝑍 ↑m π‘Œ) ↔ 𝐺:π‘ŒβŸΆπ‘))
3533, 34mpbird 256 . 2 (πœ‘ β†’ 𝐺 ∈ (𝑍 ↑m π‘Œ))
36 setcco.f . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
379, 8elmapd 8737 . . 3 (πœ‘ β†’ (𝐹 ∈ (π‘Œ ↑m 𝑋) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
3836, 37mpbird 256 . 2 (πœ‘ β†’ 𝐹 ∈ (π‘Œ ↑m 𝑋))
39 coexg 7858 . . 3 ((𝐺 ∈ (𝑍 ↑m π‘Œ) ∧ 𝐹 ∈ (π‘Œ ↑m 𝑋)) β†’ (𝐺 ∘ 𝐹) ∈ V)
4035, 38, 39syl2anc 584 . 2 (πœ‘ β†’ (𝐺 ∘ 𝐹) ∈ V)
4129, 32, 35, 38, 40ovmpod 7501 1 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3443  βŸ¨cop 4590   Γ— cxp 5629   ∘ ccom 5635  βŸΆwf 6489  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  1st c1st 7911  2nd c2nd 7912   ↑m cmap 8723  compcco 17105  SetCatcsetc 17921
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-er 8606  df-map 8725  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-dec 12577  df-uz 12722  df-fz 13379  df-struct 16979  df-slot 17014  df-ndx 17026  df-base 17044  df-hom 17117  df-cco 17118  df-setc 17922
This theorem is referenced by:  setccatid  17930  setcmon  17933  setcepi  17934  setcsect  17935  resssetc  17938  funcestrcsetclem9  17996  funcsetcestrclem9  18011  hofcllem  18107  yonedalem4c  18126  yonedalem3b  18128  yonedainv  18130  funcringcsetcALTV2lem9  46243  funcringcsetclem9ALTV  46266
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