MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stfval Structured version   Visualization version   GIF version

Theorem 1stfval 18236
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 6919 . . . . . . 7 (Base‘𝑐) ∈ V
5 fvex 6919 . . . . . . 7 (Base‘𝑑) ∈ V
64, 5xpex 7773 . . . . . 6 ((Base‘𝑐) × (Base‘𝑑)) ∈ V
76a1i 11 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8 simpl 482 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6910 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 simpr 484 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1110fveq2d 6910 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
129, 11xpeq12d 5716 . . . . . 6 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13 1stfval.t . . . . . . . 8 𝑇 = (𝐶 ×c 𝐷)
14 eqid 2737 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2737 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
1613, 14, 15xpcbas 18223 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17 1stfval.b . . . . . . 7 𝐵 = (Base‘𝑇)
1816, 17eqtr4i 2768 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
1912, 18eqtrdi 2793 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = 𝐵)
20 simpr 484 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2120reseq2d 5997 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st𝑏) = (1st𝐵))
22 simpll 767 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23 simplr 769 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
2422, 23oveq12d 7449 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
2524, 13eqtr4di 2795 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
2625fveq2d 6910 . . . . . . . . . 10 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27 1stfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝑇)
2826, 27eqtr4di 2795 . . . . . . . . 9 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
2928oveqd 7448 . . . . . . . 8 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
3029reseq2d 5997 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (1st ↾ (𝑥𝐻𝑦)))
3120, 20, 30mpoeq123dv 7508 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
3221, 31opeq12d 4881 . . . . 5 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
337, 19, 32csbied2 3936 . . . 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 18218 . . . 4 1stF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ((Base‘𝑐) × (Base‘𝑑)) / 𝑏⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
35 opex 5469 . . . 4 ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ ∈ V
3633, 34, 35ovmpoa 7588 . . 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, 37eqtrid 2789 1 (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  cop 4632   × cxp 5683  cres 5687  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  Basecbs 17247  Hom chom 17308  Catccat 17707   ×c cxpc 18213   1stF c1stf 18214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-xpc 18217  df-1stf 18218
This theorem is referenced by:  1stf1  18237  1stf2  18238  1stfcl  18242
  Copyright terms: Public domain W3C validator