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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 18232 | . . . . . . . 8 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))) |
6 | 5 | fveq2d 6901 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))) |
7 | 6 | fveq1d 6899 | . . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)) |
8 | 1, 7 | eqtrid 2780 | . . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)) |
9 | 8 | fveq2d 6901 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))) |
10 | 9 | oveqd 7437 | . . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)) |
11 | 10 | fveq1d 6899 | . 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹)) |
12 | eqid 2728 | . . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) | |
13 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
14 | eqid 2728 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
15 | eqid 2728 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
16 | 14, 3, 4, 15 | 1stfcl 18188 | . . 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 2728 | . . 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 18219 | . 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 7423 | . . . 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 18169 | . . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
28 | eqid 2728 | . . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
29 | 18, 20 | opelxpd 5717 | . . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) |
30 | 18, 23 | opelxpd 5717 | . . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵)) |
31 | 14, 27, 28, 3, 4, 15, 29, 30 | 1stf2 18184 | . . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))) |
32 | 31 | fveq1d 6899 | . . . 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 2780 | . . 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 2728 | . . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
35 | 13, 34, 22, 3, 18 | catidcl 17662 | . . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
36 | 35, 24 | opelxpd 5717 | . . . . 5 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
37 | 14, 13, 17, 34, 21, 18, 20, 18, 23, 28 | xpchom2 18177 | . . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
38 | 36, 37 | eleqtrrd 2832 | . . . 4 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)) |
39 | 38 | fvresd 6917 | . . 3 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩)) |
40 | op1stg 8005 | . . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋)) | |
41 | 35, 24, 40 | syl2anc 583 | . . 3 ⊢ (𝜑 → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋)) |
42 | 33, 39, 41 | 3eqtrd 2772 | . 2 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1 ‘𝑋)) |
43 | 11, 25, 42 | 3eqtrd 2772 | 1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⟨cop 4635 × cxp 5676 ↾ cres 5680 ‘cfv 6548 (class class class)co 7420 1st c1st 7991 2nd c2nd 7992 Basecbs 17180 Hom chom 17244 Catccat 17644 Idccid 17645 ×c cxpc 18159 1stF c1stf 18160 curryF ccurf 18202 Δfunccdiag 18204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-hom 17257 df-cco 17258 df-cat 17648 df-cid 17649 df-func 17844 df-xpc 18163 df-1stf 18164 df-curf 18206 df-diag 18208 |
This theorem is referenced by: curf2ndf 18239 |
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