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Theorem diag2 17483
Description: Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
diag2.l 𝐿 = (𝐶Δfunc𝐷)
diag2.a 𝐴 = (Base‘𝐶)
diag2.b 𝐵 = (Base‘𝐷)
diag2.h 𝐻 = (Hom ‘𝐶)
diag2.c (𝜑𝐶 ∈ Cat)
diag2.d (𝜑𝐷 ∈ Cat)
diag2.x (𝜑𝑋𝐴)
diag2.y (𝜑𝑌𝐴)
diag2.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
diag2 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Proof of Theorem diag2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 diag2.l . . . . . 6 𝐿 = (𝐶Δfunc𝐷)
2 diag2.c . . . . . 6 (𝜑𝐶 ∈ Cat)
3 diag2.d . . . . . 6 (𝜑𝐷 ∈ Cat)
41, 2, 3diagval 17478 . . . . 5 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
54fveq2d 6667 . . . 4 (𝜑 → (2nd𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
65oveqd 7162 . . 3 (𝜑 → (𝑋(2nd𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
76fveq1d 6665 . 2 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8 eqid 2818 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9 diag2.a . . 3 𝐴 = (Base‘𝐶)
10 eqid 2818 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 eqid 2818 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1210, 2, 3, 111stfcl 17435 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
13 diag2.b . . 3 𝐵 = (Base‘𝐷)
14 diag2.h . . 3 𝐻 = (Hom ‘𝐶)
15 eqid 2818 . . 3 (Id‘𝐷) = (Id‘𝐷)
16 diag2.x . . 3 (𝜑𝑋𝐴)
17 diag2.y . . 3 (𝜑𝑌𝐴)
18 diag2.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
19 eqid 2818 . . 3 ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹)
208, 9, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19curf2 17467 . 2 (𝜑 → ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹) = (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))))
2110, 9, 13xpcbas 17416 . . . . . . 7 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22 eqid 2818 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
232adantr 481 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐶 ∈ Cat)
243adantr 481 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
25 opelxpi 5585 . . . . . . . 8 ((𝑋𝐴𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
2616, 25sylan 580 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27 opelxpi 5585 . . . . . . . 8 ((𝑌𝐴𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2817, 27sylan 580 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2910, 21, 22, 23, 24, 11, 26, 281stf2 17431 . . . . . 6 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
3029oveqd 7162 . . . . 5 ((𝜑𝑥𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)))
31 df-ov 7148 . . . . . 6 (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩)
3218adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝐹 ∈ (𝑋𝐻𝑌))
33 eqid 2818 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
34 simpr 485 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝑥𝐵)
3513, 33, 15, 24, 34catidcl 16941 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
3632, 35opelxpd 5586 . . . . . . . 8 ((𝜑𝑥𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
3716adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋𝐴)
3817adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑌𝐴)
3910, 9, 13, 14, 33, 37, 34, 38, 34, 22xpchom2 17424 . . . . . . . 8 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
4036, 39eleqtrrd 2913 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
4140fvresd 6683 . . . . . 6 ((𝜑𝑥𝐵) → ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
4231, 41syl5eq 2865 . . . . 5 ((𝜑𝑥𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43 op1stg 7690 . . . . . 6 ((𝐹 ∈ (𝑋𝐻𝑌) ∧ ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥)) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4418, 35, 43syl2an2r 681 . . . . 5 ((𝜑𝑥𝐵) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4530, 42, 443eqtrd 2857 . . . 4 ((𝜑𝑥𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
4645mpteq2dva 5152 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥𝐵𝐹))
47 fconstmpt 5607 . . 3 (𝐵 × {𝐹}) = (𝑥𝐵𝐹)
4846, 47syl6eqr 2871 . 2 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
497, 20, 483eqtrd 2857 1 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {csn 4557  cop 4563  cmpt 5137   × cxp 5546  cres 5550  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Basecbs 16471  Hom chom 16564  Catccat 16923  Idccid 16924   ×c cxpc 17406   1stF c1stf 17407   curryF ccurf 17448  Δfunccdiag 17450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-func 17116  df-xpc 17410  df-1stf 17411  df-curf 17452  df-diag 17454
This theorem is referenced by:  diag2cl  17484
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