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Theorem diag12 18207
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 18203 . . . . . . . 8 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
65fveq2d 6888 . . . . . . 7 (𝜑 → (1st𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
76fveq1d 6886 . . . . . 6 (𝜑 → ((1st𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
81, 7eqtrid 2778 . . . . 5 (𝜑𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
98fveq2d 6888 . . . 4 (𝜑 → (2nd𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
109oveqd 7421 . . 3 (𝜑 → (𝑌(2nd𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
1110fveq1d 6886 . 2 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12 eqid 2726 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13 diag11.a . . 3 𝐴 = (Base‘𝐶)
14 eqid 2726 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15 eqid 2726 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1614, 3, 4, 151stfcl 18159 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
17 diag11.b . . 3 𝐵 = (Base‘𝐷)
18 diag11.c . . 3 (𝜑𝑋𝐴)
19 eqid 2726 . . 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 18190 . 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 7407 . . . 4 (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩)
2714, 13, 17xpcbas 18140 . . . . . 6 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28 eqid 2726 . . . . . 6 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
2918, 20opelxpd 5708 . . . . . 6 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3018, 23opelxpd 5708 . . . . . 6 (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
3114, 27, 28, 3, 4, 15, 29, 301stf2 18155 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
3231fveq1d 6886 . . . 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 2778 . . 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 2726 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3513, 34, 22, 3, 18catidcl 17633 . . . . . 6 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
3635, 24opelxpd 5708 . . . . 5 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3714, 13, 17, 34, 21, 18, 20, 18, 23, 28xpchom2 18148 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3836, 37eleqtrrd 2830 . . . 4 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
3938fvresd 6904 . . 3 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩) = (1st ‘⟨( 1𝑋), 𝐹⟩))
40 op1stg 7983 . . . 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 2770 . 2 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1𝑋))
4311, 25, 423eqtrd 2770 1 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cop 4629   × cxp 5667  cres 5671  cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  Basecbs 17151  Hom chom 17215  Catccat 17615  Idccid 17616   ×c cxpc 18130   1stF c1stf 18131   curryF ccurf 18173  Δfunccdiag 18175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  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  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-hom 17228  df-cco 17229  df-cat 17619  df-cid 17620  df-func 17815  df-xpc 18134  df-1stf 18135  df-curf 18177  df-diag 18179
This theorem is referenced by:  curf2ndf  18210
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