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Theorem diag11 18200

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 18197	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))
6	5	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐿) = (1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷))))
7	6	fveq1d 6829	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐿)‘𝑋) = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
8	1, 7	eqtrid 2786	. . . 4 ⊢ (𝜑 → 𝐾 = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
9	8	fveq2d 6831	. . 3 ⊢ (𝜑 → (1^st ‘𝐾) = (1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋)))
10	9	fveq1d 6829	. 2 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = ((1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))‘𝑌))
11		eqid 2739	. . 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 2739	. . . 4 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
14		eqid 2739	. . . 4 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
15	13, 3, 4, 14	1stfcl 18154	. . 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 2739	. . 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 18183	. 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 7359	. . . 4 ⊢ (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)
22	13, 12, 16	xpcbas 18135	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
23		eqid 2739	. . . . 5 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
24	17, 19	opelxpd 5657	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
25	13, 22, 23, 3, 4, 14, 24	1stf1 18149	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
26	21, 25	eqtrid 2786	. . 3 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = (1^st ‘⟨𝑋, 𝑌⟩))
27		op1stg 7943	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	17, 19, 27	syl2anc 590	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
29	26, 28	eqtrd 2774	. 2 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = 𝑋)
30	10, 20, 29	3eqtrd 2778	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 ⟨cop 4561 × cxp 5616 ‘cfv 6485 (class class class)co 7356 1^st c1st 7929 Basecbs 17170 Hom chom 17222 Catccat 17621 ×_c cxpc 18125 1^st_F c1stf 18126 curry_F ccurf 18167 Δ_funccdiag 18169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-hom 17235 df-cco 17236 df-cat 17625 df-cid 17626 df-func 17816 df-xpc 18129 df-1stf 18130 df-curf 18171 df-diag 18173

This theorem is referenced by: curf2ndf 18204 diag1 49794 prcofdiag1 49883 oppfdiag1 49904 isinito2lem 49988 isinito3 49990 diag2f1olem 50026 concl 50151 coccl 50152 concom 50153 coccom 50154 islmd 50155 iscmd 50156

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