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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 17484 | . . . . . . . 8 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
6 | 5 | fveq2d 6669 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))) |
7 | 6 | fveq1d 6667 | . . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
8 | 1, 7 | syl5eq 2868 | . . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
9 | 8 | fveq2d 6669 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))) |
10 | 9 | oveqd 7167 | . . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)) |
11 | 10 | fveq1d 6667 | . 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹)) |
12 | eqid 2821 | . . 3 ⊢ (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) | |
13 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
14 | eqid 2821 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
15 | eqid 2821 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
16 | 14, 3, 4, 15 | 1stfcl 17441 | . . 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 2821 | . . 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 17471 | . 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 7153 | . . . 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 17422 | . . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
28 | eqid 2821 | . . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
29 | 18, 20 | opelxpd 5588 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) |
30 | 18, 23 | opelxpd 5588 | . . . . . 6 ⊢ (𝜑 → 〈𝑋, 𝑍〉 ∈ (𝐴 × 𝐵)) |
31 | 14, 27, 28, 3, 4, 15, 29, 30 | 1stf2 17437 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉) = (1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))) |
32 | 31 | fveq1d 6667 | . . . 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 2868 | . . 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 2821 | . . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
35 | 13, 34, 22, 3, 18 | catidcl 16947 | . . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
36 | 35, 24 | opelxpd 5588 | . . . . 5 ⊢ (𝜑 → 〈( 1 ‘𝑋), 𝐹〉 ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
37 | 14, 13, 17, 34, 21, 18, 20, 18, 23, 28 | xpchom2 17430 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
38 | 36, 37 | eleqtrrd 2916 | . . . 4 ⊢ (𝜑 → 〈( 1 ‘𝑋), 𝐹〉 ∈ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉)) |
39 | 38 | fvresd 6685 | . . 3 ⊢ (𝜑 → ((1st ↾ (〈𝑋, 𝑌〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑍〉))‘〈( 1 ‘𝑋), 𝐹〉) = (1st ‘〈( 1 ‘𝑋), 𝐹〉)) |
40 | op1stg 7695 | . . . 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 2860 | . 2 ⊢ (𝜑 → (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘(𝐶 1stF 𝐷))〈𝑋, 𝑍〉)𝐹) = ( 1 ‘𝑋)) |
43 | 11, 25, 42 | 3eqtrd 2860 | 1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4567 × cxp 5548 ↾ cres 5552 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 Basecbs 16477 Hom chom 16570 Catccat 16929 Idccid 16930 ×c cxpc 17412 1stF c1stf 17413 curryF ccurf 17454 Δfunccdiag 17456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-hom 16583 df-cco 16584 df-cat 16933 df-cid 16934 df-func 17122 df-xpc 17416 df-1stf 17417 df-curf 17458 df-diag 17460 |
This theorem is referenced by: curf2ndf 17491 |
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