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Theorem 2ndfval 17027
Description: Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
1stfval.t 𝑇 = (𝐶 ×c 𝐷)
1stfval.b 𝐵 = (Base‘𝑇)
1stfval.h 𝐻 = (Hom ‘𝑇)
1stfval.c (𝜑𝐶 ∈ Cat)
1stfval.d (𝜑𝐷 ∈ Cat)
2ndfval.p 𝑄 = (𝐶 2ndF 𝐷)
Assertion
Ref Expression
2ndfval (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑄(𝑥,𝑦)   𝑇(𝑥,𝑦)

Proof of Theorem 2ndfval
Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2ndfval.p . 2 𝑄 = (𝐶 2ndF 𝐷)
2 1stfval.c . . 3 (𝜑𝐶 ∈ Cat)
3 1stfval.d . . 3 (𝜑𝐷 ∈ Cat)
4 fvex 6354 . . . . . . 7 (Base‘𝑐) ∈ V
5 fvex 6354 . . . . . . 7 (Base‘𝑑) ∈ V
64, 5xpex 7119 . . . . . 6 ((Base‘𝑐) × (Base‘𝑑)) ∈ V
76a1i 11 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8 simpl 474 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6348 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 simpr 479 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1110fveq2d 6348 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
129, 11xpeq12d 5289 . . . . . 6 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13 1stfval.t . . . . . . . 8 𝑇 = (𝐶 ×c 𝐷)
14 eqid 2752 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2752 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
1613, 14, 15xpcbas 17011 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17 1stfval.b . . . . . . 7 𝐵 = (Base‘𝑇)
1816, 17eqtr4i 2777 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
1912, 18syl6eq 2802 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = 𝐵)
20 simpr 479 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2120reseq2d 5543 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (2nd𝑏) = (2nd𝐵))
22 simpll 807 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23 simplr 809 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
2422, 23oveq12d 6823 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
2524, 13syl6eqr 2804 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
2625fveq2d 6348 . . . . . . . . . 10 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27 1stfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝑇)
2826, 27syl6eqr 2804 . . . . . . . . 9 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
2928oveqd 6822 . . . . . . . 8 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
3029reseq2d 5543 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (2nd ↾ (𝑥𝐻𝑦)))
3120, 20, 30mpt2eq123dv 6874 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
3221, 31opeq12d 4553 . . . . 5 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
337, 19, 32csbied2 3694 . . . 4 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
34 df-2ndf 17007 . . . 4 2ndF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ((Base‘𝑐) × (Base‘𝑑)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
35 opex 5073 . . . 4 ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ ∈ V
3633, 34, 35ovmpt2a 6948 . . 3 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 2ndF 𝐷) = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
372, 3, 36syl2anc 696 . 2 (𝜑 → (𝐶 2ndF 𝐷) = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
381, 37syl5eq 2798 1 (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  csb 3666  cop 4319   × cxp 5256  cres 5260  cfv 6041  (class class class)co 6805  cmpt2 6807  2nd c2nd 7324  Basecbs 16051  Hom chom 16146  Catccat 16518   ×c cxpc 17001   2ndF c2ndf 17003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-hom 16160  df-cco 16161  df-xpc 17005  df-2ndf 17007
This theorem is referenced by:  2ndf1  17028  2ndf2  17029  2ndfcl  17031
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