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Theorem diag11 18424

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 = (𝐶Δ_func𝐷)
diagval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diagval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
diag11.a	⊢ 𝐴 = (Base‘𝐶)
diag11.c	⊢ (𝜑 → 𝑋 ∈ 𝐴)
diag11.k	⊢ 𝐾 = ((1^st ‘𝐿)‘𝑋)
diag11.b	⊢ 𝐵 = (Base‘𝐷)
diag11.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

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

**Proof of Theorem diag11**
Step	Hyp	Ref	Expression
1		diag11.k	. . . . 5 ⊢ 𝐾 = ((1^st ‘𝐿)‘𝑋)
2		diagval.l	. . . . . . . 8 ⊢ 𝐿 = (𝐶Δ_func𝐷)
3		diagval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		diagval.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
5	2, 3, 4	diagval 18421	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))
6	5	fveq2d 6892	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐿) = (1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷))))
7	6	fveq1d 6890	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐿)‘𝑋) = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
8	1, 7	eqtrid 2807	. . . 4 ⊢ (𝜑 → 𝐾 = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
9	8	fveq2d 6892	. . 3 ⊢ (𝜑 → (1^st ‘𝐾) = (1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋)))
10	9	fveq1d 6890	. 2 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = ((1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))‘𝑌))
11		eqid 2760	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)) = (⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷))
12		diag11.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
13		eqid 2760	. . . 4 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
14		eqid 2760	. . . 4 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
15	13, 3, 4, 14	1stfcl 18378	. . 3 ⊢ (𝜑 → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
16		diag11.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
17		diag11.c	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18		eqid 2760	. . 3 ⊢ ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋) = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋)
19		diag11.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	11, 12, 3, 4, 15, 16, 17, 18, 19	curf11 18407	. 2 ⊢ (𝜑 → ((1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))‘𝑌) = (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌))
21		df-ov 7425	. . . 4 ⊢ (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)
22	13, 12, 16	xpcbas 18359	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
23		eqid 2760	. . . . 5 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
24	17, 19	opelxpd 5689	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
25	13, 22, 23, 3, 4, 14, 24	1stf1 18373	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
26	21, 25	eqtrid 2807	. . 3 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = (1^st ‘⟨𝑋, 𝑌⟩))
27		op1stg 8015	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	17, 19, 27	syl2anc 596	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
29	26, 28	eqtrd 2795	. 2 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = 𝑋)
30	10, 20, 29	3eqtrd 2799	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 × cxp 5648 ‘cfv 6541 (class class class)co 7422 1^st c1st 8001 Basecbs 17394 Hom chom 17446 Catccat 17845 ×_c cxpc 18349 1^st_F c1stf 18350 curry_F ccurf 18391 Δ_funccdiag 18393

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-er 8722 df-map 8854 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-z 12699 df-dec 12820 df-uz 12971 df-fz 13645 df-struct 17332 df-slot 17367 df-ndx 17379 df-base 17395 df-hom 17459 df-cco 17460 df-cat 17849 df-cid 17850 df-func 18040 df-xpc 18353 df-1stf 18354 df-curf 18395 df-diag 18397

This theorem is used by: curf2ndf 18428 diag1 50533 prcofdiag1 50635 oppfdiag1 50656 isinito2lem 50740 isinito3 50742 diag2f1olem 50781 par2diag 50887 concl 50951 coccl 50952 concom 50953 coccom 50954 islmd 50955 iscmd 50956

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