Theorem diag12 18210

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 18206	. . . . . . . 8 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
6	5	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
7	6	fveq1d 6842	. . . . . 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 6844	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
10	9	oveqd 7384	. . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
11	10	fveq1d 6842	. 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 18163	. . 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 18193	. 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 7370	. . . 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 18144	. . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28		eqid 2736	. . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
29	18, 20	opelxpd 5670	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	18, 23	opelxpd 5670	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
31	14, 27, 28, 3, 4, 15, 29, 30	1stf2 18159	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
32	31	fveq1d 6842	. . . 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 17648	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
36	35, 24	opelxpd 5670	. . . . 5 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
37	14, 13, 17, 34, 21, 18, 20, 18, 23, 28	xpchom2 18152	. . . . 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 6860	. . 3 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩))
40		op1stg 7954	. . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋))
41	35, 24, 40	syl2anc 585	. . 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 1542   ∈ wcel 2114  ⟨cop 4573   × cxp 5629   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Hom chom 17231  Catccat 17630  Idccid 17631   ×_c cxpc 18134   1^st_F c1stf 18135   curry_F ccurf 18176  Δ_funccdiag 18178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17634  df-cid 17635  df-func 17825  df-xpc 18138  df-1stf 18139  df-curf 18180  df-diag 18182

This theorem is referenced by:  curf2ndf  18213  diag1  49779  prcofdiag1  49868  oppfdiag1  49889  isinito2lem  49973  concom  50138  coccom  50139