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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 18134 | . . . . . . . 8 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))) |
6 | 5 | fveq2d 6847 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))) |
7 | 6 | fveq1d 6845 | . . . . . 6 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)) |
8 | 1, 7 | eqtrid 2785 | . . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)) |
9 | 8 | fveq2d 6847 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))) |
10 | 9 | oveqd 7375 | . . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)) |
11 | 10 | fveq1d 6845 | . 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹)) |
12 | eqid 2733 | . . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) | |
13 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
14 | eqid 2733 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
15 | eqid 2733 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
16 | 14, 3, 4, 15 | 1stfcl 18090 | . . 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 2733 | . . 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 18121 | . 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 7361 | . . . 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 18071 | . . . . . 6 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
28 | eqid 2733 | . . . . . 6 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
29 | 18, 20 | opelxpd 5672 | . . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵)) |
30 | 18, 23 | opelxpd 5672 | . . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵)) |
31 | 14, 27, 28, 3, 4, 15, 29, 30 | 1stf2 18086 | . . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))) |
32 | 31 | fveq1d 6845 | . . . 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 2785 | . . 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 2733 | . . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
35 | 13, 34, 22, 3, 18 | catidcl 17567 | . . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
36 | 35, 24 | opelxpd 5672 | . . . . 5 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
37 | 14, 13, 17, 34, 21, 18, 20, 18, 23, 28 | xpchom2 18079 | . . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍))) |
38 | 36, 37 | eleqtrrd 2837 | . . . 4 ⊢ (𝜑 → ⟨( 1 ‘𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)) |
39 | 38 | fvresd 6863 | . . 3 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1 ‘𝑋), 𝐹⟩) = (1st ‘⟨( 1 ‘𝑋), 𝐹⟩)) |
40 | op1stg 7934 | . . . 4 ⊢ ((( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋)) | |
41 | 35, 24, 40 | syl2anc 585 | . . 3 ⊢ (𝜑 → (1st ‘⟨( 1 ‘𝑋), 𝐹⟩) = ( 1 ‘𝑋)) |
42 | 33, 39, 41 | 3eqtrd 2777 | . 2 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1 ‘𝑋)) |
43 | 11, 25, 42 | 3eqtrd 2777 | 1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 × cxp 5632 ↾ cres 5636 ‘cfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 Basecbs 17088 Hom chom 17149 Catccat 17549 Idccid 17550 ×c cxpc 18061 1stF c1stf 18062 curryF ccurf 18104 Δfunccdiag 18106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-hom 17162 df-cco 17163 df-cat 17553 df-cid 17554 df-func 17749 df-xpc 18065 df-1stf 18066 df-curf 18108 df-diag 18110 |
This theorem is referenced by: curf2ndf 18141 |
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