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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 18285 | . . . . . . . 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 2789 | . . . . 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 7448 | . . 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 2737 | . . 3 ⊢ (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) | |
| 13 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
| 14 | eqid 2737 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
| 15 | eqid 2737 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
| 16 | 14, 3, 4, 15 | 1stfcl 18242 | . . 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 2737 | . . 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 18272 | . 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 7434 | . . . 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 18223 | . . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
| 28 | eqid 2737 | . . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
| 29 | 18, 20 | opelxpd 5724 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) |
| 30 | 18, 23 | opelxpd 5724 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑍〉 ∈ (𝐴 × 𝐵)) |
| 31 | 14, 27, 28, 3, 4, 15, 29, 30 | 1stf2 18238 | . . . . 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 2789 | . . 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 2737 | . . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 35 | 13, 34, 22, 3, 18 | catidcl 17725 | . . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
| 36 | 35, 24 | opelxpd 5724 | . . . . 5 ⊢ (𝜑 → 〈( 1 ‘𝑋), 𝐹〉 ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
| 37 | 14, 13, 17, 34, 21, 18, 20, 18, 23, 28 | xpchom2 18231 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
| 38 | 36, 37 | eleqtrrd 2844 | . . . 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 8026 | . . . 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 2781 | . 2 ⊢ (𝜑 → (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)𝐹) = ( 1 ‘𝑋)) |
| 43 | 11, 25, 42 | 3eqtrd 2781 | 1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 Basecbs 17247 Hom chom 17308 Catccat 17707 Idccid 17708 ×c cxpc 18213 1stF c1stf 18214 curryF ccurf 18255 Δfunccdiag 18257 |
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-hom 17321 df-cco 17322 df-cat 17711 df-cid 17712 df-func 17903 df-xpc 18217 df-1stf 18218 df-curf 18259 df-diag 18261 |
| This theorem is referenced by: curf2ndf 18292 diag1 49004 |
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