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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 17492 | . . . . . . . 8 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
6 | 5 | fveq2d 6676 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))) |
7 | 6 | fveq1d 6674 | . . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
8 | 1, 7 | syl5eq 2870 | . . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
9 | 8 | fveq2d 6676 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))) |
10 | 9 | oveqd 7175 | . . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)) |
11 | 10 | fveq1d 6674 | . 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹)) |
12 | eqid 2823 | . . 3 ⊢ (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) | |
13 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
14 | eqid 2823 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
15 | eqid 2823 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
16 | 14, 3, 4, 15 | 1stfcl 17449 | . . 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 2823 | . . 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 17479 | . 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 7161 | . . . 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 17430 | . . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
28 | eqid 2823 | . . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
29 | 18, 20 | opelxpd 5595 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) |
30 | 18, 23 | opelxpd 5595 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑍〉 ∈ (𝐴 × 𝐵)) |
31 | 14, 27, 28, 3, 4, 15, 29, 30 | 1stf2 17445 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉) = (1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))) |
32 | 31 | fveq1d 6674 | . . . 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 | syl5eq 2870 | . . 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 2823 | . . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
35 | 13, 34, 22, 3, 18 | catidcl 16955 | . . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
36 | 35, 24 | opelxpd 5595 | . . . . 5 ⊢ (𝜑 → 〈( 1 ‘𝑋), 𝐹〉 ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
37 | 14, 13, 17, 34, 21, 18, 20, 18, 23, 28 | xpchom2 17438 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
38 | 36, 37 | eleqtrrd 2918 | . . . 4 ⊢ (𝜑 → 〈( 1 ‘𝑋), 𝐹〉 ∈ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉)) |
39 | 38 | fvresd 6692 | . . 3 ⊢ (𝜑 → ((1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))‘〈( 1 ‘𝑋), 𝐹〉) = (1st ‘〈( 1 ‘𝑋), 𝐹〉)) |
40 | op1stg 7703 | . . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘〈( 1 ‘𝑋), 𝐹〉) = ( 1 ‘𝑋)) | |
41 | 35, 24, 40 | syl2anc 586 | . . 3 ⊢ (𝜑 → (1st ‘〈( 1 ‘𝑋), 𝐹〉) = ( 1 ‘𝑋)) |
42 | 33, 39, 41 | 3eqtrd 2862 | . 2 ⊢ (𝜑 → (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)𝐹) = ( 1 ‘𝑋)) |
43 | 11, 25, 42 | 3eqtrd 2862 | 1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 Basecbs 16485 Hom chom 16578 Catccat 16937 Idccid 16938 ×c cxpc 17420 1stF c1stf 17421 curryF ccurf 17462 Δfunccdiag 17464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-hom 16591 df-cco 16592 df-cat 16941 df-cid 16942 df-func 17130 df-xpc 17424 df-1stf 17425 df-curf 17466 df-diag 17468 |
This theorem is referenced by: curf2ndf 17499 |
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