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Mirrors > Home > MPE Home > Th. List > diag11 | Structured version Visualization version GIF version |
Description: Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-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) |
Ref | Expression |
---|---|
diag11 | ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diag11.k | . . . . 5 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) | |
2 | diagval.l | . . . . . . . 8 ⊢ 𝐿 = (𝐶Δfunc𝐷) | |
3 | diagval.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | diagval.d | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
5 | 2, 3, 4 | diagval 16880 | . . . . . . 7 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
6 | 5 | fveq2d 6195 | . . . . . 6 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))) |
7 | 6 | fveq1d 6193 | . . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
8 | 1, 7 | syl5eq 2668 | . . . 4 ⊢ (𝜑 → 𝐾 = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
9 | 8 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (1st ‘𝐾) = (1st ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))) |
10 | 9 | fveq1d 6193 | . 2 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = ((1st ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌)) |
11 | eqid 2622 | . . 3 ⊢ (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) | |
12 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
13 | eqid 2622 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
14 | eqid 2622 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
15 | 13, 3, 4, 14 | 1stfcl 16837 | . . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) |
16 | diag11.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
17 | diag11.c | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
18 | eqid 2622 | . . 3 ⊢ ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋) | |
19 | diag11.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | 11, 12, 3, 4, 15, 16, 17, 18, 19 | curf11 16866 | . 2 ⊢ (𝜑 → ((1st ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌) = (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌)) |
21 | df-ov 6653 | . . . 4 ⊢ (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = ((1st ‘(𝐶 1stF 𝐷))‘〈𝑋, 𝑌〉) | |
22 | 13, 12, 16 | xpcbas 16818 | . . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
23 | eqid 2622 | . . . . 5 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
24 | opelxpi 5148 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) | |
25 | 17, 19, 24 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) |
26 | 13, 22, 23, 3, 4, 14, 25 | 1stf1 16832 | . . . 4 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘〈𝑋, 𝑌〉) = (1st ‘〈𝑋, 𝑌〉)) |
27 | 21, 26 | syl5eq 2668 | . . 3 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = (1st ‘〈𝑋, 𝑌〉)) |
28 | op1stg 7180 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
29 | 17, 19, 28 | syl2anc 693 | . . 3 ⊢ (𝜑 → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
30 | 27, 29 | eqtrd 2656 | . 2 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = 𝑋) |
31 | 10, 20, 30 | 3eqtrd 2660 | 1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 〈cop 4183 × cxp 5112 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 Basecbs 15857 Hom chom 15952 Catccat 16325 ×c cxpc 16808 1stF c1stf 16809 curryF ccurf 16850 Δfunccdiag 16852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-hom 15966 df-cco 15967 df-cat 16329 df-cid 16330 df-func 16518 df-xpc 16812 df-1stf 16813 df-curf 16854 df-diag 16856 |
This theorem is referenced by: curf2ndf 16887 |
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