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Theorem diag11 18288
Description: Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-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)
Assertion
Ref Expression
diag11 (𝜑 → ((1st𝐾)‘𝑌) = 𝑋)

Proof of Theorem diag11
StepHypRef Expression
1 diag11.k . . . . 5 𝐾 = ((1st𝐿)‘𝑋)
2 diagval.l . . . . . . . 8 𝐿 = (𝐶Δfunc𝐷)
3 diagval.c . . . . . . . 8 (𝜑𝐶 ∈ Cat)
4 diagval.d . . . . . . . 8 (𝜑𝐷 ∈ Cat)
52, 3, 4diagval 18285 . . . . . . 7 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
65fveq2d 6910 . . . . . 6 (𝜑 → (1st𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
76fveq1d 6908 . . . . 5 (𝜑 → ((1st𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
81, 7eqtrid 2789 . . . 4 (𝜑𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
98fveq2d 6910 . . 3 (𝜑 → (1st𝐾) = (1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
109fveq1d 6908 . 2 (𝜑 → ((1st𝐾)‘𝑌) = ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌))
11 eqid 2737 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
12 diag11.a . . 3 𝐴 = (Base‘𝐶)
13 eqid 2737 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
14 eqid 2737 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1513, 3, 4, 141stfcl 18242 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
16 diag11.b . . 3 𝐵 = (Base‘𝐷)
17 diag11.c . . 3 (𝜑𝑋𝐴)
18 eqid 2737 . . 3 ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)
19 diag11.y . . 3 (𝜑𝑌𝐵)
2011, 12, 3, 4, 15, 16, 17, 18, 19curf11 18271 . 2 (𝜑 → ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌) = (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌))
21 df-ov 7434 . . . 4 (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)
2213, 12, 16xpcbas 18223 . . . . 5 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
23 eqid 2737 . . . . 5 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
2417, 19opelxpd 5724 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
2513, 22, 23, 3, 4, 14, 241stf1 18237 . . . 4 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
2621, 25eqtrid 2789 . . 3 (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = (1st ‘⟨𝑋, 𝑌⟩))
27 op1stg 8026 . . . 4 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2817, 19, 27syl2anc 584 . . 3 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2926, 28eqtrd 2777 . 2 (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = 𝑋)
3010, 20, 293eqtrd 2781 1 (𝜑 → ((1st𝐾)‘𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cop 4632   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  Basecbs 17247  Hom chom 17308  Catccat 17707   ×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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