Theorem diag12 18318

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 18314	. . . . . . . 8 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
6	5	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
7	6	fveq1d 6887	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
8	1, 7	eqtrid 2812	. . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
9	8	fveq2d 6889	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
10	9	oveqd 7433	. . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
11	10	fveq1d 6887	. 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12		eqid 2765	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13		diag11.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
14		eqid 2765	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15		eqid 2765	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
16	14, 3, 4, 15	1stfcl 18271	. . 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 2765	. . 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 18301	. 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 7419	. . . 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 18252	. . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28		eqid 2765	. . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
29	18, 20	opelxpd 5702	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	18, 23	opelxpd 5702	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
31	14, 27, 28, 3, 4, 15, 29, 30	1stf2 18267	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
32	31	fveq1d 6887	. . . 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 2812	. . 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 2765	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
35	13, 34, 22, 3, 18	catidcl 17756	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
36	35, 24	opelxpd 5702	. . . . 5 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
37	14, 13, 17, 34, 21, 18, 20, 18, 23, 28	xpchom2 18260	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
38	36, 37	eleqtrrd 2868	. . . 4 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
39	38	fvresd 6905	. . 3 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩))
40		op1stg 8000	. . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋))
41	35, 24, 40	syl2anc 596	. . 3 ⊢ (𝜑 → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋))
42	33, 39, 41	3eqtrd 2804	. 2 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1 ‘𝑋))
43	11, 25, 42	3eqtrd 2804	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cop 4597   × cxp 5661   ↾ cres 5665  ‘cfv 6540  (class class class)co 7416  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17287  Hom chom 17339  Catccat 17738  Idccid 17739   ×_c cxpc 18242   1^st_F c1stf 18243   curry_F ccurf 18284  Δ_funccdiag 18286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-struct 17225  df-slot 17260  df-ndx 17272  df-base 17288  df-hom 17352  df-cco 17353  df-cat 17742  df-cid 17743  df-func 17933  df-xpc 18246  df-1stf 18247  df-curf 18288  df-diag 18290

This theorem is used by:  curf2ndf  18321  diag1  50115  prcofdiag1  50204  oppfdiag1  50225  isinito2lem  50309  concom  50474  coccom  50475