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Theorem diag12 18289
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 18285 . . . . . . . 8 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
65fveq2d 6910 . . . . . . 7 (𝜑 → (1st𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
76fveq1d 6908 . . . . . 6 (𝜑 → ((1st𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
81, 7eqtrid 2789 . . . . 5 (𝜑𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
98fveq2d 6910 . . . 4 (𝜑 → (2nd𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
109oveqd 7448 . . 3 (𝜑 → (𝑌(2nd𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
1110fveq1d 6908 . 2 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12 eqid 2737 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13 diag11.a . . 3 𝐴 = (Base‘𝐶)
14 eqid 2737 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15 eqid 2737 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1614, 3, 4, 151stfcl 18242 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
17 diag11.b . . 3 𝐵 = (Base‘𝐷)
18 diag11.c . . 3 (𝜑𝑋𝐴)
19 eqid 2737 . . 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 18272 . 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 7434 . . . 4 (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩)
2714, 13, 17xpcbas 18223 . . . . . 6 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28 eqid 2737 . . . . . 6 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
2918, 20opelxpd 5724 . . . . . 6 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3018, 23opelxpd 5724 . . . . . 6 (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
3114, 27, 28, 3, 4, 15, 29, 301stf2 18238 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
3231fveq1d 6908 . . . 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 2789 . . 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 2737 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3513, 34, 22, 3, 18catidcl 17725 . . . . . 6 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
3635, 24opelxpd 5724 . . . . 5 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3714, 13, 17, 34, 21, 18, 20, 18, 23, 28xpchom2 18231 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3836, 37eleqtrrd 2844 . . . 4 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
3938fvresd 6926 . . 3 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩) = (1st ‘⟨( 1𝑋), 𝐹⟩))
40 op1stg 8026 . . . 4 ((( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4135, 24, 40syl2anc 584 . . 3 (𝜑 → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4233, 39, 413eqtrd 2781 . 2 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1𝑋))
4311, 25, 423eqtrd 2781 1 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cop 4632   × cxp 5683  cres 5687  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  Catccat 17707  Idccid 17708   ×c cxpc 18213   1stF c1stf 18214   curryF ccurf 18255  Δfunccdiag 18257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-func 17903  df-xpc 18217  df-1stf 18218  df-curf 18259  df-diag 18261
This theorem is referenced by:  curf2ndf  18292  diag1  49004
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