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Theorem diag12 16707
 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 16703 . . . . . . . 8 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
65fveq2d 6107 . . . . . . 7 (𝜑 → (1st𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
76fveq1d 6105 . . . . . 6 (𝜑 → ((1st𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
81, 7syl5eq 2656 . . . . 5 (𝜑𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
98fveq2d 6107 . . . 4 (𝜑 → (2nd𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
109oveqd 6566 . . 3 (𝜑 → (𝑌(2nd𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
1110fveq1d 6105 . 2 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12 eqid 2610 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13 diag11.a . . 3 𝐴 = (Base‘𝐶)
14 eqid 2610 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15 eqid 2610 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1614, 3, 4, 151stfcl 16660 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
17 diag11.b . . 3 𝐵 = (Base‘𝐷)
18 diag11.c . . 3 (𝜑𝑋𝐴)
19 eqid 2610 . . 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 16690 . 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 6552 . . . 4 (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩)
2714, 13, 17xpcbas 16641 . . . . . 6 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28 eqid 2610 . . . . . 6 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
29 opelxpi 5072 . . . . . . 7 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3018, 20, 29syl2anc 691 . . . . . 6 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
31 opelxpi 5072 . . . . . . 7 ((𝑋𝐴𝑍𝐵) → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
3218, 23, 31syl2anc 691 . . . . . 6 (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
3314, 27, 28, 3, 4, 15, 30, 321stf2 16656 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
3433fveq1d 6105 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩))
3526, 34syl5eq 2656 . . 3 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩))
36 eqid 2610 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3713, 36, 22, 3, 18catidcl 16166 . . . . . 6 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
38 opelxpi 5072 . . . . . 6 ((( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → ⟨( 1𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3937, 24, 38syl2anc 691 . . . . 5 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
4014, 13, 17, 36, 21, 18, 20, 18, 23, 28xpchom2 16649 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
4139, 40eleqtrrd 2691 . . . 4 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
42 fvres 6117 . . . 4 (⟨( 1𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩) = (1st ‘⟨( 1𝑋), 𝐹⟩))
4341, 42syl 17 . . 3 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩) = (1st ‘⟨( 1𝑋), 𝐹⟩))
44 op1stg 7071 . . . 4 ((( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4537, 24, 44syl2anc 691 . . 3 (𝜑 → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4635, 43, 453eqtrd 2648 . 2 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1𝑋))
4711, 25, 463eqtrd 2648 1 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  Catccat 16148  Idccid 16149   ×c cxpc 16631   1stF c1stf 16632   curryF ccurf 16673  Δfunccdiag 16675 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-func 16341  df-xpc 16635  df-1stf 16636  df-curf 16677  df-diag 16679 This theorem is referenced by:  curf2ndf  16710
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