Theorem diag2 18151

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 18146	. . . . 5 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
5	4	fveq2d 6826	. . . 4 ⊢ (𝜑 → (2nd ‘𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
6	5	oveqd 7363	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
7	6	fveq1d 6824	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8		eqid 2731	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9		diag2.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
10		eqid 2731	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11		eqid 2731	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
12	10, 2, 3, 11	1stfcl 18103	. . 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 2731	. . 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 2731	. . 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 18135	. 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 18084	. . . . . . 7 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22		eqid 2731	. . . . . . 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 5653	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
26	16, 25	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27		opelxpi 5653	. . . . . . . 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 18099	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
30	29	oveqd 7363	. . . . 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 7349	. . . . . 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 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
34		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35	13, 33, 15, 24, 34	catidcl 17588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
36	32, 35	opelxpd 5655	. . . . . . . 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 18092	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
40	36, 39	eleqtrrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
41	40	fvresd 6842	. . . . . 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 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43		op1stg 7933	. . . . . 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 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
46	45	mpteq2dva 5184	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥 ∈ 𝐵 ↦ 𝐹))
47		fconstmpt 5678	. . 3 ⊢ (𝐵 × {𝐹}) = (𝑥 ∈ 𝐵 ↦ 𝐹)
48	46, 47	eqtr4di 2784	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
49	7, 20, 48	3eqtrd 2770	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {csn 4576  ⟨cop 4582   ↦ cmpt 5172   × cxp 5614   ↾ cres 5618  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17120  Hom chom 17172  Catccat 17570  Idccid 17571   ×_c cxpc 18074   1^st_F c1stf 18075   curry_F ccurf 18116  Δ_funccdiag 18118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-func 17765  df-xpc 18078  df-1stf 18079  df-curf 18120  df-diag 18122

This theorem is referenced by:  diag2cl  18152  diag2f1lem  49346  prcofdiag  49432  oppfdiag  49454  diag2f1olem  49574  islmd  49703  iscmd  49704