Theorem diag2 18290

Description: Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
diag2.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
diag2.a	⊢ 𝐴 = (Base‘𝐶)
diag2.b	⊢ 𝐵 = (Base‘𝐷)
diag2.h	⊢ 𝐻 = (Hom ‘𝐶)
diag2.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diag2.d	⊢ (𝜑 → 𝐷 ∈ Cat)
diag2.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
diag2.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
diag2.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
diag2	⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Proof of Theorem diag2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diag2.l	. . . . . 6 ⊢ 𝐿 = (𝐶Δfunc𝐷)
2		diag2.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		diag2.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
4	1, 2, 3	diagval 18285	. . . . 5 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
5	4	fveq2d 6910	. . . 4 ⊢ (𝜑 → (2nd ‘𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
6	5	oveqd 7448	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
7	6	fveq1d 6908	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8		eqid 2737	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9		diag2.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
10		eqid 2737	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11		eqid 2737	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
12	10, 2, 3, 11	1stfcl 18242	. . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
13		diag2.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
14		diag2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
15		eqid 2737	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
16		diag2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
17		diag2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
18		diag2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
19		eqid 2737	. . 3 ⊢ ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹)
20	8, 9, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19	curf2 18274	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))))
21	10, 9, 13	xpcbas 18223	. . . . . . 7 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22		eqid 2737	. . . . . . 7 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
23	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ Cat)
24	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
25		opelxpi 5722	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
26	16, 25	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27		opelxpi 5722	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
28	17, 27	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
29	10, 21, 22, 23, 24, 11, 26, 28	1stf2 18238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
30	29	oveqd 7448	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)))
31		df-ov 7434	. . . . . 6 ⊢ (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩)
32	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝑋𝐻𝑌))
33		eqid 2737	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
34		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35	13, 33, 15, 24, 34	catidcl 17725	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
36	32, 35	opelxpd 5724	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
37	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐴)
38	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑌 ∈ 𝐴)
39	10, 9, 13, 14, 33, 37, 34, 38, 34, 22	xpchom2 18231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
40	36, 39	eleqtrrd 2844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
41	40	fvresd 6926	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
42	31, 41	eqtrid 2789	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43		op1stg 8026	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥)) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
44	18, 35, 43	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
45	30, 42, 44	3eqtrd 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
46	45	mpteq2dva 5242	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥 ∈ 𝐵 ↦ 𝐹))
47		fconstmpt 5747	. . 3 ⊢ (𝐵 × {𝐹}) = (𝑥 ∈ 𝐵 ↦ 𝐹)
48	46, 47	eqtr4di 2795	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
49	7, 20, 48	3eqtrd 2781	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {csn 4626  ⟨cop 4632   ↦ cmpt 5225   × cxp 5683   ↾ cres 5687  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  Basecbs 17247  Hom chom 17308  Catccat 17707  Idccid 17708   ×_c cxpc 18213   1^st_F c1stf 18214   curry_F ccurf 18255  Δ_funccdiag 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:  diag2cl  18291