MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ndfval Structured version   Visualization version   GIF version

Theorem 2ndfval 18192
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 6915 . . . . . . 7 (Base‘𝑐) ∈ V
5 fvex 6915 . . . . . . 7 (Base‘𝑑) ∈ V
64, 5xpex 7761 . . . . . 6 ((Base‘𝑐) × (Base‘𝑑)) ∈ V
76a1i 11 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8 simpl 481 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6906 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 simpr 483 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1110fveq2d 6906 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
129, 11xpeq12d 5713 . . . . . 6 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13 1stfval.t . . . . . . . 8 𝑇 = (𝐶 ×c 𝐷)
14 eqid 2728 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2728 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
1613, 14, 15xpcbas 18176 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17 1stfval.b . . . . . . 7 𝐵 = (Base‘𝑇)
1816, 17eqtr4i 2759 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
1912, 18eqtrdi 2784 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = 𝐵)
20 simpr 483 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2120reseq2d 5989 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (2nd𝑏) = (2nd𝐵))
22 simpll 765 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23 simplr 767 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
2422, 23oveq12d 7444 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
2524, 13eqtr4di 2786 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
2625fveq2d 6906 . . . . . . . . . 10 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27 1stfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝑇)
2826, 27eqtr4di 2786 . . . . . . . . 9 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
2928oveqd 7443 . . . . . . . 8 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
3029reseq2d 5989 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (2nd ↾ (𝑥𝐻𝑦)))
3120, 20, 30mpoeq123dv 7501 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
3221, 31opeq12d 4886 . . . . 5 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
337, 19, 32csbied2 3934 . . . 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 18172 . . . 4 2ndF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ((Base‘𝑐) × (Base‘𝑑)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
35 opex 5470 . . . 4 ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ ∈ V
3633, 34, 35ovmpoa 7582 . . 3 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 2ndF 𝐷) = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
372, 3, 36syl2anc 582 . 2 (𝜑 → (𝐶 2ndF 𝐷) = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
381, 37eqtrid 2780 1 (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  csb 3894  cop 4638   × cxp 5680  cres 5684  cfv 6553  (class class class)co 7426  cmpo 7428  2nd c2nd 7998  Basecbs 17187  Hom chom 17251  Catccat 17651   ×c cxpc 18166   2ndF c2ndf 18168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-slot 17158  df-ndx 17170  df-base 17188  df-hom 17264  df-cco 17265  df-xpc 18170  df-2ndf 18172
This theorem is referenced by:  2ndf1  18193  2ndf2  18194  2ndfcl  18196
  Copyright terms: Public domain W3C validator