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Theorem diag2 18194
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 18189 . . . . 5 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
54fveq2d 6892 . . . 4 (𝜑 → (2nd𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
65oveqd 7422 . . 3 (𝜑 → (𝑋(2nd𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
76fveq1d 6890 . 2 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8 eqid 2732 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9 diag2.a . . 3 𝐴 = (Base‘𝐶)
10 eqid 2732 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 eqid 2732 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1210, 2, 3, 111stfcl 18145 . . 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 2732 . . 3 (Id‘𝐷) = (Id‘𝐷)
16 diag2.x . . 3 (𝜑𝑋𝐴)
17 diag2.y . . 3 (𝜑𝑌𝐴)
18 diag2.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
19 eqid 2732 . . 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 18178 . 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 18126 . . . . . . 7 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22 eqid 2732 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
232adantr 481 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐶 ∈ Cat)
243adantr 481 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
25 opelxpi 5712 . . . . . . . 8 ((𝑋𝐴𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
2616, 25sylan 580 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27 opelxpi 5712 . . . . . . . 8 ((𝑌𝐴𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2817, 27sylan 580 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2910, 21, 22, 23, 24, 11, 26, 281stf2 18141 . . . . . 6 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
3029oveqd 7422 . . . . 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 7408 . . . . . 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 2732 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
34 simpr 485 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝑥𝐵)
3513, 33, 15, 24, 34catidcl 17622 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
3632, 35opelxpd 5713 . . . . . . . 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 18134 . . . . . . . 8 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
4036, 39eleqtrrd 2836 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
4140fvresd 6908 . . . . . 6 ((𝜑𝑥𝐵) → ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
4231, 41eqtrid 2784 . . . . 5 ((𝜑𝑥𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43 op1stg 7983 . . . . . 6 ((𝐹 ∈ (𝑋𝐻𝑌) ∧ ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥)) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4418, 35, 43syl2an2r 683 . . . . 5 ((𝜑𝑥𝐵) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4530, 42, 443eqtrd 2776 . . . 4 ((𝜑𝑥𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
4645mpteq2dva 5247 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥𝐵𝐹))
47 fconstmpt 5736 . . 3 (𝐵 × {𝐹}) = (𝑥𝐵𝐹)
4846, 47eqtr4di 2790 . 2 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
497, 20, 483eqtrd 2776 1 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {csn 4627  cop 4633  cmpt 5230   × cxp 5673  cres 5677  cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  Basecbs 17140  Hom chom 17204  Catccat 17604  Idccid 17605   ×c cxpc 18116   1stF c1stf 18117   curryF ccurf 18159  Δfunccdiag 18161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17141  df-hom 17217  df-cco 17218  df-cat 17608  df-cid 17609  df-func 17804  df-xpc 18120  df-1stf 18121  df-curf 18163  df-diag 18165
This theorem is referenced by:  diag2cl  18195
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