Theorem diag12 16707

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 16703	. . . . . . . 8 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
6	5	fveq2d 6107	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
7	6	fveq1d 6105	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
8	1, 7	syl5eq 2656	. . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
9	8	fveq2d 6107	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
10	9	oveqd 6566	. . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
11	10	fveq1d 6105	. 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹))
12		eqid 2610	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
13		diag11.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
14		eqid 2610	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
15		eqid 2610	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
16	14, 3, 4, 15	1stfcl 16660	. . 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 2610	. . 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 16690	. 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 6552	. . . 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 16641	. . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28		eqid 2610	. . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
29		opelxpi 5072	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
30	18, 20, 29	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
31		opelxpi 5072	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑍 ∈ 𝐵) → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
32	18, 23, 31	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
33	14, 27, 28, 3, 4, 15, 30, 32	1stf2 16656	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
34	33	fveq1d 6105	. . . 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⟩))
35	26, 34	syl5eq 2656	. . 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⟩))
36		eqid 2610	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	13, 36, 22, 3, 18	catidcl 16166	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
38		opelxpi 5072	. . . . . 6 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
39	37, 24, 38	syl2anc 691	. . . . 5 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
40	14, 13, 17, 36, 21, 18, 20, 18, 23, 28	xpchom2 16649	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
41	39, 40	eleqtrrd 2691	. . . 4 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
42		fvres 6117	. . . 4 ⊢ (⟨( 1 ‘𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩))
43	41, 42	syl 17	. . 3 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩))
44		op1stg 7071	. . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋))
45	37, 24, 44	syl2anc 691	. . 3 ⊢ (𝜑 → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋))
46	35, 43, 45	3eqtrd 2648	. 2 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1 ‘𝑋))
47	11, 25, 46	3eqtrd 2648	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  Basecbs 15695  Hom chom 15779  Catccat 16148  Idccid 16149   ×_c cxpc 16631   1^st_F c1stf 16632   curry_F ccurf 16673  Δ_funccdiag 16675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-func 16341  df-xpc 16635  df-1stf 16636  df-curf 16677  df-diag 16679

This theorem is referenced by:  curf2ndf  16710