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Theorem diag2 16708
 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 16703 . . . . 5 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
54fveq2d 6107 . . . 4 (𝜑 → (2nd𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
65oveqd 6566 . . 3 (𝜑 → (𝑋(2nd𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
76fveq1d 6105 . 2 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8 eqid 2610 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9 diag2.a . . 3 𝐴 = (Base‘𝐶)
10 eqid 2610 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 eqid 2610 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1210, 2, 3, 111stfcl 16660 . . 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 2610 . . 3 (Id‘𝐷) = (Id‘𝐷)
16 diag2.x . . 3 (𝜑𝑋𝐴)
17 diag2.y . . 3 (𝜑𝑌𝐴)
18 diag2.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
19 eqid 2610 . . 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 16692 . 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 16641 . . . . . . 7 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22 eqid 2610 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
232adantr 480 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐶 ∈ Cat)
243adantr 480 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
25 opelxpi 5072 . . . . . . . 8 ((𝑋𝐴𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
2616, 25sylan 487 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27 opelxpi 5072 . . . . . . . 8 ((𝑌𝐴𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2817, 27sylan 487 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2910, 21, 22, 23, 24, 11, 26, 281stf2 16656 . . . . . 6 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
3029oveqd 6566 . . . . 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 6552 . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝐹 ∈ (𝑋𝐻𝑌))
33 eqid 2610 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
34 simpr 476 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝑥𝐵)
3513, 33, 15, 24, 34catidcl 16166 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
36 opelxpi 5072 . . . . . . . . 9 ((𝐹 ∈ (𝑋𝐻𝑌) ∧ ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥)) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
3732, 35, 36syl2anc 691 . . . . . . . 8 ((𝜑𝑥𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
3816adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋𝐴)
3917adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑌𝐴)
4010, 9, 13, 14, 33, 38, 34, 39, 34, 22xpchom2 16649 . . . . . . . 8 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
4137, 40eleqtrrd 2691 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
42 fvres 6117 . . . . . . 7 (⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) → ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
4341, 42syl 17 . . . . . 6 ((𝜑𝑥𝐵) → ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
4431, 43syl5eq 2656 . . . . 5 ((𝜑𝑥𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
45 op1stg 7071 . . . . . 6 ((𝐹 ∈ (𝑋𝐻𝑌) ∧ ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥)) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4632, 35, 45syl2anc 691 . . . . 5 ((𝜑𝑥𝐵) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4730, 44, 463eqtrd 2648 . . . 4 ((𝜑𝑥𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
4847mpteq2dva 4672 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥𝐵𝐹))
49 fconstmpt 5085 . . 3 (𝐵 × {𝐹}) = (𝑥𝐵𝐹)
5048, 49syl6eqr 2662 . 2 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
517, 20, 503eqtrd 2648 1 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  Catccat 16148  Idccid 16149   ×c cxpc 16631   1stF c1stf 16632   curryF ccurf 16673  Δfunccdiag 16675 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-func 16341  df-xpc 16635  df-1stf 16636  df-curf 16677  df-diag 16679 This theorem is referenced by:  diag2cl  16709
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