Theorem diag12 18254

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 18250	. . . . . . . 8 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
6	5	fveq2d 6879	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
7	6	fveq1d 6877	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
8	1, 7	eqtrid 2782	. . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
9	8	fveq2d 6879	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
10	9	oveqd 7420	. . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
11	10	fveq1d 6877	. 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12		eqid 2735	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13		diag11.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
14		eqid 2735	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15		eqid 2735	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
16	14, 3, 4, 15	1stfcl 18207	. . 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 2735	. . 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 18237	. 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 7406	. . . 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 18188	. . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28		eqid 2735	. . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
29	18, 20	opelxpd 5693	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	18, 23	opelxpd 5693	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
31	14, 27, 28, 3, 4, 15, 29, 30	1stf2 18203	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
32	31	fveq1d 6877	. . . 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 2782	. . 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 2735	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
35	13, 34, 22, 3, 18	catidcl 17692	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
36	35, 24	opelxpd 5693	. . . . 5 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
37	14, 13, 17, 34, 21, 18, 20, 18, 23, 28	xpchom2 18196	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
38	36, 37	eleqtrrd 2837	. . . 4 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
39	38	fvresd 6895	. . 3 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩))
40		op1stg 7998	. . . 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 2774	. 2 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1 ‘𝑋))
43	11, 25, 42	3eqtrd 2774	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ⟨cop 4607   × cxp 5652   ↾ cres 5656  ‘cfv 6530  (class class class)co 7403  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17226  Hom chom 17280  Catccat 17674  Idccid 17675   ×_c cxpc 18178   1^st_F c1stf 18179   curry_F ccurf 18220  Δ_funccdiag 18222

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-map 8840  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-fz 13523  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-hom 17293  df-cco 17294  df-cat 17678  df-cid 17679  df-func 17869  df-xpc 18182  df-1stf 18183  df-curf 18224  df-diag 18226

This theorem is referenced by:  curf2ndf  18257  diag1  49163  isinito2lem  49331  concom  49481  coccom  49482