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Theorem diag12 18138
Description: Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
diagval.l 𝐿 = (𝐶Δfunc𝐷)
diagval.c (𝜑𝐶 ∈ Cat)
diagval.d (𝜑𝐷 ∈ Cat)
diag11.a 𝐴 = (Base‘𝐶)
diag11.c (𝜑𝑋𝐴)
diag11.k 𝐾 = ((1st𝐿)‘𝑋)
diag11.b 𝐵 = (Base‘𝐷)
diag11.y (𝜑𝑌𝐵)
diag12.j 𝐽 = (Hom ‘𝐷)
diag12.i 1 = (Id‘𝐶)
diag12.z (𝜑𝑍𝐵)
diag12.f (𝜑𝐹 ∈ (𝑌𝐽𝑍))
Assertion
Ref Expression
diag12 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))

Proof of Theorem diag12
StepHypRef Expression
1 diag11.k . . . . . 6 𝐾 = ((1st𝐿)‘𝑋)
2 diagval.l . . . . . . . . 9 𝐿 = (𝐶Δfunc𝐷)
3 diagval.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
4 diagval.d . . . . . . . . 9 (𝜑𝐷 ∈ Cat)
52, 3, 4diagval 18134 . . . . . . . 8 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
65fveq2d 6847 . . . . . . 7 (𝜑 → (1st𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
76fveq1d 6845 . . . . . 6 (𝜑 → ((1st𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
81, 7eqtrid 2785 . . . . 5 (𝜑𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
98fveq2d 6847 . . . 4 (𝜑 → (2nd𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
109oveqd 7375 . . 3 (𝜑 → (𝑌(2nd𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍))
1110fveq1d 6845 . 2 (𝜑 → ((𝑌(2nd𝐾)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)
1614, 3, 4, 151stfcl 18090 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
17 diag11.b . . 3 𝐵 = (Base‘𝐷)
18 diag11.c . . 3 (𝜑𝑋𝐴)
19 eqid 2733 . . 3 ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)
20 diag11.y . . 3 (𝜑𝑌𝐵)
21 diag12.j . . 3 𝐽 = (Hom ‘𝐷)
22 diag12.i . . 3 1 = (Id‘𝐶)
23 diag12.z . . 3 (𝜑𝑍𝐵)
24 diag12.f . . 3 (𝜑𝐹 ∈ (𝑌𝐽𝑍))
2512, 13, 3, 4, 16, 17, 18, 19, 20, 21, 22, 23, 24curf12 18121 . 2 (𝜑 → ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))𝑍)‘𝐹) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹))
26 df-ov 7361 . . . 4 (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩)
2714, 13, 17xpcbas 18071 . . . . . 6 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
28 eqid 2733 . . . . . 6 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
2918, 20opelxpd 5672 . . . . . 6 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3018, 23opelxpd 5672 . . . . . 6 (𝜑 → ⟨𝑋, 𝑍⟩ ∈ (𝐴 × 𝐵))
3114, 27, 28, 3, 4, 15, 29, 301stf2 18086 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩)))
3231fveq1d 6845 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)‘⟨( 1𝑋), 𝐹⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩))
3326, 32eqtrid 2785 . . 3 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩))
34 eqid 2733 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3513, 34, 22, 3, 18catidcl 17567 . . . . . 6 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
3635, 24opelxpd 5672 . . . . 5 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3714, 13, 17, 34, 21, 18, 20, 18, 23, 28xpchom2 18079 . . . . 5 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑌𝐽𝑍)))
3836, 37eleqtrrd 2837 . . . 4 (𝜑 → ⟨( 1𝑋), 𝐹⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))
3938fvresd 6863 . . 3 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑍⟩))‘⟨( 1𝑋), 𝐹⟩) = (1st ‘⟨( 1𝑋), 𝐹⟩))
40 op1stg 7934 . . . 4 ((( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑌𝐽𝑍)) → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4135, 24, 40syl2anc 585 . . 3 (𝜑 → (1st ‘⟨( 1𝑋), 𝐹⟩) = ( 1𝑋))
4233, 39, 413eqtrd 2777 . 2 (𝜑 → (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑋, 𝑍⟩)𝐹) = ( 1𝑋))
4311, 25, 423eqtrd 2777 1 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))
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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