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Theorem diag12 18236
Description: Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
diagval.l 𝐿 = (𝐶Δfunc𝐷)
diagval.c (𝜑𝐶 ∈ Cat)
diagval.d (𝜑𝐷 ∈ Cat)
diag11.a 𝐴 = (Base‘𝐶)
diag11.c (𝜑𝑋𝐴)
diag11.k 𝐾 = ((1st𝐿)‘𝑋)
diag11.b 𝐵 = (Base‘𝐷)
diag11.y (𝜑𝑌𝐵)
diag12.j 𝐽 = (Hom ‘𝐷)
diag12.i 1 = (Id‘𝐶)
diag12.z (𝜑𝑍𝐵)
diag12.f (𝜑𝐹 ∈ (𝑌𝐽𝑍))
Assertion
Ref Expression
diag12 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))

Proof of Theorem diag12
StepHypRef Expression
1 diag11.k . . . . . 6 𝐾 = ((1st𝐿)‘𝑋)
2 diagval.l . . . . . . . . 9 𝐿 = (𝐶Δfunc𝐷)
3 diagval.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
4 diagval.d . . . . . . . . 9 (𝜑𝐷 ∈ Cat)
52, 3, 4diagval 18232 . . . . . . . 8 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
65fveq2d 6901 . . . . . . 7 (𝜑 → (1st𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
76fveq1d 6899 . . . . . 6 (𝜑 → ((1st𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
81, 7eqtrid 2780 . . . . 5 (𝜑𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
98fveq2d 6901 . . . 4 (𝜑 → (2nd𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
109oveqd 7437 . . 3 (𝜑 → (𝑌(2nd𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
1110fveq1d 6899 . 2 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12 eqid 2728 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13 diag11.a . . 3 𝐴 = (Base‘𝐶)
14 eqid 2728 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15 eqid 2728 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1614, 3, 4, 151stfcl 18188 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
17 diag11.b . . 3 𝐵 = (Base‘𝐷)
18 diag11.c . . 3 (𝜑𝑋𝐴)
19 eqid 2728 . . 3 ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)
20 diag11.y . . 3 (𝜑𝑌𝐵)
21 diag12.j . . 3 𝐽 = (Hom ‘𝐷)
22 diag12.i . . 3 1 = (Id‘𝐶)
23 diag12.z . . 3 (𝜑𝑍𝐵)
24 diag12.f . . 3 (𝜑𝐹 ∈ (𝑌𝐽𝑍))
2512, 13, 3, 4, 16, 17, 18, 19, 20, 21, 22, 23, 24curf12 18219 . 2 (𝜑 → ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹))
26 df-ov 7423 . . . 4 (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩)
2714, 13, 17xpcbas 18169 . . . . . 6 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28 eqid 2728 . . . . . 6 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
2918, 20opelxpd 5717 . . . . . 6 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3018, 23opelxpd 5717 . . . . . 6 (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
3114, 27, 28, 3, 4, 15, 29, 301stf2 18184 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
3231fveq1d 6899 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩))
3326, 32eqtrid 2780 . . 3 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩))
34 eqid 2728 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3513, 34, 22, 3, 18catidcl 17662 . . . . . 6 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
3635, 24opelxpd 5717 . . . . 5 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3714, 13, 17, 34, 21, 18, 20, 18, 23, 28xpchom2 18177 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3836, 37eleqtrrd 2832 . . . 4 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
3938fvresd 6917 . . 3 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩) = (1st ‘⟨( 1𝑋), 𝐹⟩))
40 op1stg 8005 . . . 4 ((( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4135, 24, 40syl2anc 583 . . 3 (𝜑 → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4233, 39, 413eqtrd 2772 . 2 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1𝑋))
4311, 25, 423eqtrd 2772 1 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cop 4635   × cxp 5676  cres 5680  cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  Basecbs 17180  Hom chom 17244  Catccat 17644  Idccid 17645   ×c cxpc 18159   1stF c1stf 18160   curryF ccurf 18202  Δfunccdiag 18204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-fz 13518  df-struct 17116  df-slot 17151  df-ndx 17163  df-base 17181  df-hom 17257  df-cco 17258  df-cat 17648  df-cid 17649  df-func 17844  df-xpc 18163  df-1stf 18164  df-curf 18206  df-diag 18208
This theorem is referenced by:  curf2ndf  18239
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