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Theorem diag11 18266

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 18263	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))
6	5	fveq2d 6866	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐿) = (1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷))))
7	6	fveq1d 6864	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐿)‘𝑋) = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
8	1, 7	eqtrid 2808	. . . 4 ⊢ (𝜑 → 𝐾 = ((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))
9	8	fveq2d 6866	. . 3 ⊢ (𝜑 → (1^st ‘𝐾) = (1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋)))
10	9	fveq1d 6864	. 2 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = ((1^st ‘((1^st ‘(⟨𝐶, 𝐷⟩ curry_F (𝐶 1^st_F 𝐷)))‘𝑋))‘𝑌))
11		eqid 2761	. . 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 2761	. . . 4 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
14		eqid 2761	. . . 4 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
15	13, 3, 4, 14	1stfcl 18220	. . 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 2761	. . 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 18249	. 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 7394	. . . 4 ⊢ (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩)
22	13, 12, 16	xpcbas 18201	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
23		eqid 2761	. . . . 5 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
24	17, 19	opelxpd 5682	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
25	13, 22, 23, 3, 4, 14, 24	1stf1 18215	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐶 1^st_F 𝐷))‘⟨𝑋, 𝑌⟩) = (1^st ‘⟨𝑋, 𝑌⟩))
26	21, 25	eqtrid 2808	. . 3 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = (1^st ‘⟨𝑋, 𝑌⟩))
27		op1stg 7977	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	17, 19, 27	syl2anc 593	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
29	26, 28	eqtrd 2796	. 2 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶 1^st_F 𝐷))𝑌) = 𝑋)
30	10, 20, 29	3eqtrd 2800	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ⟨cop 4585 × cxp 5641 ‘cfv 6516 (class class class)co 7391 1^st c1st 7963 Basecbs 17236 Hom chom 17288 Catccat 17687 ×_c cxpc 18191 1^st_F c1stf 18192 curry_F ccurf 18233 Δ_funccdiag 18235

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-er 8672 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-fz 13507 df-struct 17174 df-slot 17209 df-ndx 17221 df-base 17237 df-hom 17301 df-cco 17302 df-cat 17691 df-cid 17692 df-func 17882 df-xpc 18195 df-1stf 18196 df-curf 18237 df-diag 18239

This theorem is referenced by: curf2ndf 18270 diag1 49886 prcofdiag1 49975 oppfdiag1 49996 isinito2lem 50080 isinito3 50082 diag2f1olem 50118 concl 50243 coccl 50244 concom 50245 coccom 50246 islmd 50247 iscmd 50248

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