Theorem diag12 18167

Description: Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
diagval.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
diagval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diagval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
diag11.a	⊢ 𝐴 = (Base‘𝐶)
diag11.c	⊢ (𝜑 → 𝑋 ∈ 𝐴)
diag11.k	⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
diag11.b	⊢ 𝐵 = (Base‘𝐷)
diag11.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
diag12.j	⊢ 𝐽 = (Hom ‘𝐷)
diag12.i	⊢ 1 = (Id‘𝐶)
diag12.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
diag12.f	⊢ (𝜑 → 𝐹 ∈ (𝑌𝐽𝑍))

Assertion

Ref	Expression
diag12	⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋))

Proof of Theorem diag12

Step	Hyp	Ref	Expression
1		diag11.k	. . . . . 6 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
2		diagval.l	. . . . . . . . 9 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3		diagval.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		diagval.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
5	2, 3, 4	diagval 18163	. . . . . . . 8 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
6	5	fveq2d 6838	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
7	6	fveq1d 6836	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
8	1, 7	eqtrid 2783	. . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
9	8	fveq2d 6838	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
10	9	oveqd 7375	. . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
11	10	fveq1d 6836	. 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12		eqid 2736	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13		diag11.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
14		eqid 2736	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15		eqid 2736	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
16	14, 3, 4, 15	1stfcl 18120	. . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
17		diag11.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
18		diag11.c	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
19		eqid 2736	. . 3 ⊢ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)
20		diag11.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
21		diag12.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
22		diag12.i	. . 3 ⊢ 1 = (Id‘𝐶)
23		diag12.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
24		diag12.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌𝐽𝑍))
25	12, 13, 3, 4, 16, 17, 18, 19, 20, 21, 22, 23, 24	curf12 18150	. 2 ⊢ (𝜑 → ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹))
26		df-ov 7361	. . . 4 ⊢ (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1 ‘𝑋), 𝐹⟩)
27	14, 13, 17	xpcbas 18101	. . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28		eqid 2736	. . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
29	18, 20	opelxpd 5663	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	18, 23	opelxpd 5663	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
31	14, 27, 28, 3, 4, 15, 29, 30	1stf2 18116	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
32	31	fveq1d 6836	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1 ‘𝑋), 𝐹⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩))
33	26, 32	eqtrid 2783	. . 3 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩))
34		eqid 2736	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
35	13, 34, 22, 3, 18	catidcl 17605	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
36	35, 24	opelxpd 5663	. . . . 5 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
37	14, 13, 17, 34, 21, 18, 20, 18, 23, 28	xpchom2 18109	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
38	36, 37	eleqtrrd 2839	. . . 4 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
39	38	fvresd 6854	. . 3 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩))
40		op1stg 7945	. . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋))
41	35, 24, 40	syl2anc 584	. . 3 ⊢ (𝜑 → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋))
42	33, 39, 41	3eqtrd 2775	. 2 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1 ‘𝑋))
43	11, 25, 42	3eqtrd 2775	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4586   × cxp 5622   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17136  Hom chom 17188  Catccat 17587  Idccid 17588   ×_c cxpc 18091   1^st_F c1stf 18092   curry_F ccurf 18133  Δ_funccdiag 18135

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-func 17782  df-xpc 18095  df-1stf 18096  df-curf 18137  df-diag 18139

This theorem is referenced by:  curf2ndf  18170  diag1  49549  prcofdiag1  49638  oppfdiag1  49659  isinito2lem  49743  concom  49908  coccom  49909