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Theorem diag11 18253

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 18250	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))
6	5	fveq2d 6879	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐿) = (1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷))))
7	6	fveq1d 6877	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐿)‘𝑋) = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
8	1, 7	eqtrid 2782	. . . 4 ⊢ (𝜑 → 𝐾 = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
9	8	fveq2d 6879	. . 3 ⊢ (𝜑 → (1^st ‘𝐾) = (1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋)))
10	9	fveq1d 6877	. 2 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = ((1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))‘𝑌))
11		eqid 2735	. . 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 2735	. . . 4 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
14		eqid 2735	. . . 4 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
15	13, 3, 4, 14	1stfcl 18207	. . 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 2735	. . 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 18236	. 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 7406	. . . 4 ⊢ (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)
22	13, 12, 16	xpcbas 18188	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
23		eqid 2735	. . . . 5 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
24	17, 19	opelxpd 5693	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
25	13, 22, 23, 3, 4, 14, 24	1stf1 18202	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
26	21, 25	eqtrid 2782	. . 3 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = (1^st ‘⟨𝑋, 𝑌⟩))
27		op1stg 7998	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	17, 19, 27	syl2anc 584	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
29	26, 28	eqtrd 2770	. 2 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = 𝑋)
30	10, 20, 29	3eqtrd 2774	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⟨cop 4607 × cxp 5652 ‘cfv 6530 (class class class)co 7403 1^st c1st 7984 Basecbs 17226 Hom chom 17280 Catccat 17674 ×_c cxpc 18178 1^st_F c1stf 18179 curry_F ccurf 18220 Δ_funccdiag 18222

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-fz 13523 df-struct 17164 df-slot 17199 df-ndx 17211 df-base 17227 df-hom 17293 df-cco 17294 df-cat 17678 df-cid 17679 df-func 17869 df-xpc 18182 df-1stf 18183 df-curf 18224 df-diag 18226

This theorem is referenced by: curf2ndf 18257 diag1 49163 isinito2lem 49331 isinito3 49333 diag2f1olem 49369 concl 49479 coccl 49480 concom 49481 coccom 49482 islmd 49483 iscmd 49484

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