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Mirrors > Home > MPE Home > Th. List > diag12 | Structured version Visualization version GIF version |
Description: Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
Ref | Expression |
---|---|
diagval.l | ⊢ 𝐿 = (𝐶Δfunc𝐷) |
diagval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
diagval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
diag11.a | ⊢ 𝐴 = (Base‘𝐶) |
diag11.c | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
diag11.k | ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) |
diag11.b | ⊢ 𝐵 = (Base‘𝐷) |
diag11.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
diag12.j | ⊢ 𝐽 = (Hom ‘𝐷) |
diag12.i | ⊢ 1 = (Id‘𝐶) |
diag12.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
diag12.f | ⊢ (𝜑 → 𝐹 ∈ (𝑌𝐽𝑍)) |
Ref | Expression |
---|---|
diag12 | ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diag11.k | . . . . . 6 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) | |
2 | diagval.l | . . . . . . . . 9 ⊢ 𝐿 = (𝐶Δfunc𝐷) | |
3 | diagval.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | diagval.d | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
5 | 2, 3, 4 | diagval 18296 | . . . . . . . 8 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
6 | 5 | fveq2d 6910 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))) |
7 | 6 | fveq1d 6908 | . . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
8 | 1, 7 | eqtrid 2786 | . . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
9 | 8 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))) |
10 | 9 | oveqd 7447 | . . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)) |
11 | 10 | fveq1d 6908 | . 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹)) |
12 | eqid 2734 | . . 3 ⊢ (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) | |
13 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
14 | eqid 2734 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
15 | eqid 2734 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
16 | 14, 3, 4, 15 | 1stfcl 18252 | . . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) |
17 | diag11.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
18 | diag11.c | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | eqid 2734 | . . 3 ⊢ ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋) | |
20 | diag11.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
21 | diag12.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
22 | diag12.i | . . 3 ⊢ 1 = (Id‘𝐶) | |
23 | diag12.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
24 | diag12.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌𝐽𝑍)) | |
25 | 12, 13, 3, 4, 16, 17, 18, 19, 20, 21, 22, 23, 24 | curf12 18283 | . 2 ⊢ (𝜑 → ((𝑌(2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹) = (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)𝐹)) |
26 | df-ov 7433 | . . . 4 ⊢ (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)𝐹) = ((〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)‘〈( 1 ‘𝑋), 𝐹〉) | |
27 | 14, 13, 17 | xpcbas 18233 | . . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
28 | eqid 2734 | . . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
29 | 18, 20 | opelxpd 5727 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) |
30 | 18, 23 | opelxpd 5727 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑍〉 ∈ (𝐴 × 𝐵)) |
31 | 14, 27, 28, 3, 4, 15, 29, 30 | 1stf2 18248 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉) = (1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))) |
32 | 31 | fveq1d 6908 | . . . 4 ⊢ (𝜑 → ((〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)‘〈( 1 ‘𝑋), 𝐹〉) = ((1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))‘〈( 1 ‘𝑋), 𝐹〉)) |
33 | 26, 32 | eqtrid 2786 | . . 3 ⊢ (𝜑 → (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)𝐹) = ((1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))‘〈( 1 ‘𝑋), 𝐹〉)) |
34 | eqid 2734 | . . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
35 | 13, 34, 22, 3, 18 | catidcl 17726 | . . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
36 | 35, 24 | opelxpd 5727 | . . . . 5 ⊢ (𝜑 → 〈( 1 ‘𝑋), 𝐹〉 ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
37 | 14, 13, 17, 34, 21, 18, 20, 18, 23, 28 | xpchom2 18241 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
38 | 36, 37 | eleqtrrd 2841 | . . . 4 ⊢ (𝜑 → 〈( 1 ‘𝑋), 𝐹〉 ∈ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉)) |
39 | 38 | fvresd 6926 | . . 3 ⊢ (𝜑 → ((1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))‘〈( 1 ‘𝑋), 𝐹〉) = (1st ‘〈( 1 ‘𝑋), 𝐹〉)) |
40 | op1stg 8024 | . . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘〈( 1 ‘𝑋), 𝐹〉) = ( 1 ‘𝑋)) | |
41 | 35, 24, 40 | syl2anc 584 | . . 3 ⊢ (𝜑 → (1st ‘〈( 1 ‘𝑋), 𝐹〉) = ( 1 ‘𝑋)) |
42 | 33, 39, 41 | 3eqtrd 2778 | . 2 ⊢ (𝜑 → (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)𝐹) = ( 1 ‘𝑋)) |
43 | 11, 25, 42 | 3eqtrd 2778 | 1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 〈cop 4636 × cxp 5686 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 Basecbs 17244 Hom chom 17308 Catccat 17708 Idccid 17709 ×c cxpc 18223 1stF c1stf 18224 curryF ccurf 18266 Δfunccdiag 18268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-hom 17321 df-cco 17322 df-cat 17712 df-cid 17713 df-func 17908 df-xpc 18227 df-1stf 18228 df-curf 18270 df-diag 18272 |
This theorem is referenced by: curf2ndf 18303 |
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