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Theorem setcco 17961
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 17960 . . 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 7930 . . . . . . . 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 2776 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = π‘Œ)
145, 13oveq12d 7371 . . . 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 7929 . . . . . . . 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 2776 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = 𝑋)
2013, 19oveq12d 7371 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) = (π‘Œ ↑m 𝑋))
21 eqidd 2737 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
2214, 20, 21mpoeq123dv 7428 . . 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 7386 . . . . 5 (𝑍 ↑m π‘Œ) ∈ V
26 ovex 7386 . . . . 5 (π‘Œ ↑m 𝑋) ∈ V
2725, 26mpoex 8008 . . . 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 7503 . 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 8775 . . 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 8775 . . 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 7862 . . 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 7503 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 7353   ∈ cmpo 7355  1st c1st 7915  2nd c2nd 7916   ↑m cmap 8761  compcco 17137  SetCatcsetc 17953
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 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  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 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-1o 8408  df-er 8644  df-map 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12410  df-z 12496  df-dec 12615  df-uz 12760  df-fz 13417  df-struct 17011  df-slot 17046  df-ndx 17058  df-base 17076  df-hom 17149  df-cco 17150  df-setc 17954
This theorem is referenced by:  setccatid  17962  setcmon  17965  setcepi  17966  setcsect  17967  resssetc  17970  funcestrcsetclem9  18028  funcsetcestrclem9  18043  hofcllem  18139  yonedalem4c  18158  yonedalem3b  18160  yonedainv  18162  funcringcsetcALTV2lem9  46274  funcringcsetclem9ALTV  46297
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