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Theorem 1stfval 17429
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)
1stfval.p 𝑃 = (𝐶 1stF 𝐷)
Assertion
Ref Expression
1stfval (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑃(𝑥,𝑦)   𝑇(𝑥,𝑦)

Proof of Theorem 1stfval
Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.p . 2 𝑃 = (𝐶 1stF 𝐷)
2 1stfval.c . . 3 (𝜑𝐶 ∈ Cat)
3 1stfval.d . . 3 (𝜑𝐷 ∈ Cat)
4 fvex 6676 . . . . . . 7 (Base‘𝑐) ∈ V
5 fvex 6676 . . . . . . 7 (Base‘𝑑) ∈ V
64, 5xpex 7465 . . . . . 6 ((Base‘𝑐) × (Base‘𝑑)) ∈ V
76a1i 11 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8 simpl 483 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6667 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 simpr 485 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1110fveq2d 6667 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
129, 11xpeq12d 5579 . . . . . 6 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13 1stfval.t . . . . . . . 8 𝑇 = (𝐶 ×c 𝐷)
14 eqid 2818 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2818 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
1613, 14, 15xpcbas 17416 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17 1stfval.b . . . . . . 7 𝐵 = (Base‘𝑇)
1816, 17eqtr4i 2844 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
1912, 18syl6eq 2869 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = 𝐵)
20 simpr 485 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2120reseq2d 5846 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st𝑏) = (1st𝐵))
22 simpll 763 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23 simplr 765 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
2422, 23oveq12d 7163 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
2524, 13syl6eqr 2871 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
2625fveq2d 6667 . . . . . . . . . 10 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27 1stfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝑇)
2826, 27syl6eqr 2871 . . . . . . . . 9 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
2928oveqd 7162 . . . . . . . 8 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
3029reseq2d 5846 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (1st ↾ (𝑥𝐻𝑦)))
3120, 20, 30mpoeq123dv 7218 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
3221, 31opeq12d 4803 . . . . 5 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
337, 19, 32csbied2 3917 . . . 4 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) / 𝑏⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
34 df-1stf 17411 . . . 4 1stF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ((Base‘𝑐) × (Base‘𝑑)) / 𝑏⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
35 opex 5347 . . . 4 ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ ∈ V
3633, 34, 35ovmpoa 7294 . . 3 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 1stF 𝐷) = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
372, 3, 36syl2anc 584 . 2 (𝜑 → (𝐶 1stF 𝐷) = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
381, 37syl5eq 2865 1 (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  csb 3880  cop 4563   × cxp 5546  cres 5550  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  Basecbs 16471  Hom chom 16564  Catccat 16923   ×c cxpc 17406   1stF c1stf 17407
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-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-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-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  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-ndx 16474  df-slot 16475  df-base 16477  df-hom 16577  df-cco 16578  df-xpc 17410  df-1stf 17411
This theorem is referenced by:  1stf1  17430  1stf2  17431  1stfcl  17435
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