Theorem diag2 18213

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 18208	. . . . 5 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
5	4	fveq2d 6865	. . . 4 ⊢ (𝜑 → (2nd ‘𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
6	5	oveqd 7407	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
7	6	fveq1d 6863	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8		eqid 2730	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9		diag2.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
10		eqid 2730	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11		eqid 2730	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
12	10, 2, 3, 11	1stfcl 18165	. . 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 2730	. . 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 2730	. . 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 18197	. 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 18146	. . . . . . 7 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22		eqid 2730	. . . . . . 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 5678	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
26	16, 25	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27		opelxpi 5678	. . . . . . . 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 18161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
30	29	oveqd 7407	. . . . 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 7393	. . . . . 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 2730	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
34		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35	13, 33, 15, 24, 34	catidcl 17650	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
36	32, 35	opelxpd 5680	. . . . . . . 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 18154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
40	36, 39	eleqtrrd 2832	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
41	40	fvresd 6881	. . . . . 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 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43		op1stg 7983	. . . . . 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 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
46	45	mpteq2dva 5203	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥 ∈ 𝐵 ↦ 𝐹))
47		fconstmpt 5703	. . 3 ⊢ (𝐵 × {𝐹}) = (𝑥 ∈ 𝐵 ↦ 𝐹)
48	46, 47	eqtr4di 2783	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
49	7, 20, 48	3eqtrd 2769	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4592  ⟨cop 4598   ↦ cmpt 5191   × cxp 5639   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17186  Hom chom 17238  Catccat 17632  Idccid 17633   ×_c cxpc 18136   1^st_F c1stf 18137   curry_F ccurf 18178  Δ_funccdiag 18180

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-hom 17251  df-cco 17252  df-cat 17636  df-cid 17637  df-func 17827  df-xpc 18140  df-1stf 18141  df-curf 18182  df-diag 18184

This theorem is referenced by:  diag2cl  18214  diag2f1lem  49301  prcofdiag  49387  oppfdiag  49409  diag2f1olem  49529  islmd  49658  iscmd  49659