Theorem diag2 18182

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 18177	. . . . 5 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
5	4	fveq2d 6844	. . . 4 ⊢ (𝜑 → (2nd ‘𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
6	5	oveqd 7386	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
7	6	fveq1d 6842	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8		eqid 2729	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9		diag2.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
10		eqid 2729	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11		eqid 2729	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
12	10, 2, 3, 11	1stfcl 18134	. . 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 2729	. . 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 2729	. . 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 18166	. 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 18115	. . . . . . 7 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22		eqid 2729	. . . . . . 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 5668	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
26	16, 25	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27		opelxpi 5668	. . . . . . . 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 18130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
30	29	oveqd 7386	. . . . 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 7372	. . . . . 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 2729	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
34		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35	13, 33, 15, 24, 34	catidcl 17619	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
36	32, 35	opelxpd 5670	. . . . . . . 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 18123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
40	36, 39	eleqtrrd 2831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
41	40	fvresd 6860	. . . . . 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 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43		op1stg 7959	. . . . . 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 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
46	45	mpteq2dva 5195	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥 ∈ 𝐵 ↦ 𝐹))
47		fconstmpt 5693	. . 3 ⊢ (𝐵 × {𝐹}) = (𝑥 ∈ 𝐵 ↦ 𝐹)
48	46, 47	eqtr4di 2782	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
49	7, 20, 48	3eqtrd 2768	1 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4585  ⟨cop 4591   ↦ cmpt 5183   × cxp 5629   ↾ cres 5633  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  Hom chom 17207  Catccat 17601  Idccid 17602   ×_c cxpc 18105   1^st_F c1stf 18106   curry_F ccurf 18147  Δ_funccdiag 18149

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17605  df-cid 17606  df-func 17796  df-xpc 18109  df-1stf 18110  df-curf 18151  df-diag 18153

This theorem is referenced by:  diag2cl  18183  diag2f1lem  49270  prcofdiag  49356  oppfdiag  49378  diag2f1olem  49498  islmd  49627  iscmd  49628