Theorem diag2 18198

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 18193	. . . . 5 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
5	4	fveq2d 6896	. . . 4 ⊢ (𝜑 → (2nd ‘𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
6	5	oveqd 7426	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
7	6	fveq1d 6894	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8		eqid 2733	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9		diag2.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
10		eqid 2733	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11		eqid 2733	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
12	10, 2, 3, 11	1stfcl 18149	. . 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 2733	. . 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 2733	. . 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 18182	. 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 18130	. . . . . . 7 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22		eqid 2733	. . . . . . 7 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
23	2	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ Cat)
24	3	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
25		opelxpi 5714	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
26	16, 25	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27		opelxpi 5714	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
28	17, 27	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
29	10, 21, 22, 23, 24, 11, 26, 28	1stf2 18145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
30	29	oveqd 7426	. . . . 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 7412	. . . . . 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 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝑋𝐻𝑌))
33		eqid 2733	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
34		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35	13, 33, 15, 24, 34	catidcl 17626	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
36	32, 35	opelxpd 5716	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
37	16	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐴)
38	17	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑌 ∈ 𝐴)
39	10, 9, 13, 14, 33, 37, 34, 38, 34, 22	xpchom2 18138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
40	36, 39	eleqtrrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
41	40	fvresd 6912	. . . . . 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 2785	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43		op1stg 7987	. . . . . 6 ⊢ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥)) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
44	18, 35, 43	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
45	30, 42, 44	3eqtrd 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
46	45	mpteq2dva 5249	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥 ∈ 𝐵 ↦ 𝐹))
47		fconstmpt 5739	. . 3 ⊢ (𝐵 × {𝐹}) = (𝑥 ∈ 𝐵 ↦ 𝐹)
48	46, 47	eqtr4di 2791	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
49	7, 20, 48	3eqtrd 2777	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {csn 4629  ⟨cop 4635   ↦ cmpt 5232   × cxp 5675   ↾ cres 5679  ‘cfv 6544  (class class class)co 7409  1^st c1st 7973  2^nd c2nd 7974  Basecbs 17144  Hom chom 17208  Catccat 17608  Idccid 17609   ×_c cxpc 18120   1^st_F c1stf 18121   curry_F ccurf 18163  Δ_funccdiag 18165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-hom 17221  df-cco 17222  df-cat 17612  df-cid 17613  df-func 17808  df-xpc 18124  df-1stf 18125  df-curf 18167  df-diag 18169

This theorem is referenced by:  diag2cl  18199