Theorem diag11 18313

Description: Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-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)

Assertion

Ref	Expression
diag11	⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋)

Proof of Theorem diag11

Step	Hyp	Ref	Expression
1		diag11.k	. . . . 5 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
2		diagval.l	. . . . . . . 8 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3		diagval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		diagval.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
5	2, 3, 4	diagval 18310	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
6	5	fveq2d 6924	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
7	6	fveq1d 6922	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
8	1, 7	eqtrid 2792	. . . 4 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
9	8	fveq2d 6924	. . 3 ⊢ (𝜑 → (1st ‘𝐾) = (1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
10	9	fveq1d 6922	. 2 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌))
11		eqid 2740	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
12		diag11.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
13		eqid 2740	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
14		eqid 2740	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
15	13, 3, 4, 14	1stfcl 18266	. . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
16		diag11.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
17		diag11.c	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18		eqid 2740	. . 3 ⊢ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)
19		diag11.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	11, 12, 3, 4, 15, 16, 17, 18, 19	curf11 18296	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌) = (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌))
21		df-ov 7451	. . . 4 ⊢ (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)
22	13, 12, 16	xpcbas 18247	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
23		eqid 2740	. . . . 5 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
24	17, 19	opelxpd 5739	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
25	13, 22, 23, 3, 4, 14, 24	1stf1 18261	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
26	21, 25	eqtrid 2792	. . 3 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = (1st ‘⟨𝑋, 𝑌⟩))
27		op1stg 8042	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	17, 19, 27	syl2anc 583	. . 3 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
29	26, 28	eqtrd 2780	. 2 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = 𝑋)
30	10, 20, 29	3eqtrd 2784	1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ⟨cop 4654   × cxp 5698  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  Basecbs 17258  Hom chom 17322  Catccat 17722   ×_c cxpc 18237   1^st_F c1stf 18238   curry_F ccurf 18280  Δ_funccdiag 18282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-hom 17335  df-cco 17336  df-cat 17726  df-cid 17727  df-func 17922  df-xpc 18241  df-1stf 18242  df-curf 18284  df-diag 18286

This theorem is referenced by:  curf2ndf  18317