MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  setcco Structured version   Visualization version   GIF version
Theorem setcco 18033
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 18032 . . 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 772 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑧 = 𝑍)
6 simprl 770 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑣 = βŸ¨π‘‹, π‘ŒβŸ©)
76fveq2d 6896 . . . . . 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 7988 . . . . . . . 8 ((𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
118, 9, 10syl2anc 585 . . . . . . 7 (πœ‘ β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
1211adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
137, 12eqtrd 2773 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = π‘Œ)
145, 13oveq12d 7427 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑧 ↑m (2nd β€˜π‘£)) = (𝑍 ↑m π‘Œ))
156fveq2d 6896 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©))
16 op1stg 7987 . . . . . . . 8 ((𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
178, 9, 16syl2anc 585 . . . . . . 7 (πœ‘ β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
1817adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
1915, 18eqtrd 2773 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = 𝑋)
2013, 19oveq12d 7427 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) = (π‘Œ ↑m 𝑋))
21 eqidd 2734 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
2214, 20, 21mpoeq123dv 7484 . . 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 5716 . . 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 7442 . . . . 5 (𝑍 ↑m π‘Œ) ∈ V
26 ovex 7442 . . . . 5 (π‘Œ ↑m 𝑋) ∈ V
2725, 26mpoex 8066 . . . 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 7560 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍) = (𝑔 ∈ (𝑍 ↑m π‘Œ), 𝑓 ∈ (π‘Œ ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)))
30 simprl 770 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑔 = 𝐺)
31 simprr 772 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑓 = 𝐹)
3230, 31coeq12d 5865 . 2 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
33 setcco.g . . 3 (πœ‘ β†’ 𝐺:π‘ŒβŸΆπ‘)
3424, 9elmapd 8834 . . 3 (πœ‘ β†’ (𝐺 ∈ (𝑍 ↑m π‘Œ) ↔ 𝐺:π‘ŒβŸΆπ‘))
3533, 34mpbird 257 . 2 (πœ‘ β†’ 𝐺 ∈ (𝑍 ↑m π‘Œ))
36 setcco.f . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
379, 8elmapd 8834 . . 3 (πœ‘ β†’ (𝐹 ∈ (π‘Œ ↑m 𝑋) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
3836, 37mpbird 257 . 2 (πœ‘ β†’ 𝐹 ∈ (π‘Œ ↑m 𝑋))
39 coexg 7920 . . 3 ((𝐺 ∈ (𝑍 ↑m π‘Œ) ∧ 𝐹 ∈ (π‘Œ ↑m 𝑋)) β†’ (𝐺 ∘ 𝐹) ∈ V)
4035, 38, 39syl2anc 585 . 2 (πœ‘ β†’ (𝐺 ∘ 𝐹) ∈ V)
4129, 32, 35, 38, 40ovmpod 7560 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  βŸ¨cop 4635   Γ— cxp 5675   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974   ↑m cmap 8820  compcco 17209  SetCatcsetc 18025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-hom 17221  df-cco 17222  df-setc 18026
This theorem is referenced by:  setccatid  18034  setcmon  18037  setcepi  18038  setcsect  18039  resssetc  18042  funcestrcsetclem9  18100  funcsetcestrclem9  18115  hofcllem  18211  yonedalem4c  18230  yonedalem3b  18232  yonedainv  18234  funcringcsetcALTV2lem9  46942  funcringcsetclem9ALTV  46965
  Copyright terms: Public domain W3C validator