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Theorem oppccofval 17660
Description: Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
oppcco.b 𝐡 = (Baseβ€˜πΆ)
oppcco.c Β· = (compβ€˜πΆ)
oppcco.o 𝑂 = (oppCatβ€˜πΆ)
oppcco.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
oppcco.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
oppcco.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
Assertion
Ref Expression
oppccofval (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘‚)𝑍) = tpos (βŸ¨π‘, π‘ŒβŸ© Β· 𝑋))

Proof of Theorem oppccofval
Dummy variables 𝑧 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppcco.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
2 elfvex 6929 . . . . . 6 (𝑋 ∈ (Baseβ€˜πΆ) β†’ 𝐢 ∈ V)
3 oppcco.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
42, 3eleq2s 2851 . . . . 5 (𝑋 ∈ 𝐡 β†’ 𝐢 ∈ V)
5 eqid 2732 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
6 oppcco.c . . . . . 6 Β· = (compβ€˜πΆ)
7 oppcco.o . . . . . 6 𝑂 = (oppCatβ€˜πΆ)
83, 5, 6, 7oppcval 17656 . . . . 5 (𝐢 ∈ V β†’ 𝑂 = ((𝐢 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜πΆ)⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
91, 4, 83syl 18 . . . 4 (πœ‘ β†’ 𝑂 = ((𝐢 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜πΆ)⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
109fveq2d 6895 . . 3 (πœ‘ β†’ (compβ€˜π‘‚) = (compβ€˜((𝐢 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜πΆ)⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩)))
11 ovex 7441 . . . 4 (𝐢 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜πΆ)⟩) ∈ V
123fvexi 6905 . . . . . 6 𝐡 ∈ V
1312, 12xpex 7739 . . . . 5 (𝐡 Γ— 𝐡) ∈ V
1413, 12mpoex 8065 . . . 4 (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’))) ∈ V
15 ccoid 17358 . . . . 5 comp = Slot (compβ€˜ndx)
1615setsid 17140 . . . 4 (((𝐢 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜πΆ)⟩) ∈ V ∧ (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’))) ∈ V) β†’ (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’))) = (compβ€˜((𝐢 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜πΆ)⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩)))
1711, 14, 16mp2an 690 . . 3 (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’))) = (compβ€˜((𝐢 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜πΆ)⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
1810, 17eqtr4di 2790 . 2 (πœ‘ β†’ (compβ€˜π‘‚) = (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’))))
19 simprr 771 . . . . 5 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑧 = 𝑍)
20 simprl 769 . . . . . . 7 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑒 = βŸ¨π‘‹, π‘ŒβŸ©)
2120fveq2d 6895 . . . . . 6 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘’) = (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©))
22 oppcco.y . . . . . . . 8 (πœ‘ β†’ π‘Œ ∈ 𝐡)
2322adantr 481 . . . . . . 7 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ π‘Œ ∈ 𝐡)
24 op2ndg 7987 . . . . . . 7 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
251, 23, 24syl2an2r 683 . . . . . 6 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
2621, 25eqtrd 2772 . . . . 5 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘’) = π‘Œ)
2719, 26opeq12d 4881 . . . 4 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ βŸ¨π‘§, (2nd β€˜π‘’)⟩ = βŸ¨π‘, π‘ŒβŸ©)
2820fveq2d 6895 . . . . 5 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘’) = (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©))
29 op1stg 7986 . . . . . 6 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
301, 23, 29syl2an2r 683 . . . . 5 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
3128, 30eqtrd 2772 . . . 4 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘’) = 𝑋)
3227, 31oveq12d 7426 . . 3 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)) = (βŸ¨π‘, π‘ŒβŸ© Β· 𝑋))
3332tposeqd 8213 . 2 ((πœ‘ ∧ (𝑒 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)) = tpos (βŸ¨π‘, π‘ŒβŸ© Β· 𝑋))
341, 22opelxpd 5715 . 2 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝐡 Γ— 𝐡))
35 oppcco.z . 2 (πœ‘ β†’ 𝑍 ∈ 𝐡)
36 ovex 7441 . . . 4 (βŸ¨π‘, π‘ŒβŸ© Β· 𝑋) ∈ V
3736tposex 8244 . . 3 tpos (βŸ¨π‘, π‘ŒβŸ© Β· 𝑋) ∈ V
3837a1i 11 . 2 (πœ‘ β†’ tpos (βŸ¨π‘, π‘ŒβŸ© Β· 𝑋) ∈ V)
3918, 33, 34, 35, 38ovmpod 7559 1 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘‚)𝑍) = tpos (βŸ¨π‘, π‘ŒβŸ© Β· 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βŸ¨cop 4634   Γ— cxp 5674  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  1st c1st 7972  2nd c2nd 7973  tpos ctpos 8209   sSet csts 17095  ndxcnx 17125  Basecbs 17143  Hom chom 17207  compcco 17208  oppCatcoppc 17654
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  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
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-ltxr 11252  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-dec 12677  df-sets 17096  df-slot 17114  df-ndx 17126  df-cco 17221  df-oppc 17655
This theorem is referenced by:  oppcco  17661
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