Theorem diag11 18299

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 18296	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
6	5	fveq2d 6910	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
7	6	fveq1d 6908	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
8	1, 7	eqtrid 2786	. . . 4 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
9	8	fveq2d 6910	. . 3 ⊢ (𝜑 → (1st ‘𝐾) = (1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
10	9	fveq1d 6908	. 2 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌))
11		eqid 2734	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
12		diag11.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
13		eqid 2734	. . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
14		eqid 2734	. . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
15	13, 3, 4, 14	1stfcl 18252	. . 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 2734	. . 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 18282	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌) = (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌))
21		df-ov 7433	. . . 4 ⊢ (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)
22	13, 12, 16	xpcbas 18233	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
23		eqid 2734	. . . . 5 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
24	17, 19	opelxpd 5727	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
25	13, 22, 23, 3, 4, 14, 24	1stf1 18247	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
26	21, 25	eqtrid 2786	. . 3 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = (1st ‘⟨𝑋, 𝑌⟩))
27		op1stg 8024	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	17, 19, 27	syl2anc 584	. . 3 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
29	26, 28	eqtrd 2774	. 2 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = 𝑋)
30	10, 20, 29	3eqtrd 2778	1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋)

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105  ⟨cop 4636   × cxp 5686  ‘cfv 6562  (class class class)co 7430  1^st c1st 8010  Basecbs 17244  Hom chom 17308  Catccat 17708   ×_c cxpc 18223   1^st_F c1stf 18224   curry_F ccurf 18266  Δ_funccdiag 18268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-func 17908  df-xpc 18227  df-1stf 18228  df-curf 18270  df-diag 18272

This theorem is referenced by:  curf2ndf  18303