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Theorem diag11 18178

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 18175	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))
6	5	fveq2d 6846	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐿) = (1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷))))
7	6	fveq1d 6844	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐿)‘𝑋) = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
8	1, 7	eqtrid 2784	. . . 4 ⊢ (𝜑 → 𝐾 = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
9	8	fveq2d 6846	. . 3 ⊢ (𝜑 → (1^st ‘𝐾) = (1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋)))
10	9	fveq1d 6844	. 2 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = ((1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))‘𝑌))
11		eqid 2737	. . 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 2737	. . . 4 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
14		eqid 2737	. . . 4 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
15	13, 3, 4, 14	1stfcl 18132	. . 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 2737	. . 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 18161	. 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 7371	. . . 4 ⊢ (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)
22	13, 12, 16	xpcbas 18113	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
23		eqid 2737	. . . . 5 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
24	17, 19	opelxpd 5671	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
25	13, 22, 23, 3, 4, 14, 24	1stf1 18127	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
26	21, 25	eqtrid 2784	. . 3 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = (1^st ‘⟨𝑋, 𝑌⟩))
27		op1stg 7955	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	17, 19, 27	syl2anc 585	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
29	26, 28	eqtrd 2772	. 2 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = 𝑋)
30	10, 20, 29	3eqtrd 2776	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4588 × cxp 5630 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 Basecbs 17148 Hom chom 17200 Catccat 17599 ×_c cxpc 18103 1^st_F c1stf 18104 curry_F ccurf 18145 Δ_funccdiag 18147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-hom 17213 df-cco 17214 df-cat 17603 df-cid 17604 df-func 17794 df-xpc 18107 df-1stf 18108 df-curf 18149 df-diag 18151

This theorem is referenced by: curf2ndf 18182 diag1 49663 prcofdiag1 49752 oppfdiag1 49773 isinito2lem 49857 isinito3 49859 diag2f1olem 49895 concl 50020 coccl 50021 concom 50022 coccom 50023 islmd 50024 iscmd 50025

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