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Theorem 1stfval 16752
 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 6158 . . . . . . 7 (Base‘𝑐) ∈ V
5 fvex 6158 . . . . . . 7 (Base‘𝑑) ∈ V
64, 5xpex 6915 . . . . . 6 ((Base‘𝑐) × (Base‘𝑑)) ∈ V
76a1i 11 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8 simpl 473 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6152 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 simpr 477 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1110fveq2d 6152 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
129, 11xpeq12d 5100 . . . . . 6 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13 1stfval.t . . . . . . . 8 𝑇 = (𝐶 ×c 𝐷)
14 eqid 2621 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2621 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
1613, 14, 15xpcbas 16739 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17 1stfval.b . . . . . . 7 𝐵 = (Base‘𝑇)
1816, 17eqtr4i 2646 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
1912, 18syl6eq 2671 . . . . 5 ((𝑐 = 𝐶𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = 𝐵)
20 simpr 477 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2120reseq2d 5356 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st𝑏) = (1st𝐵))
22 simpll 789 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23 simplr 791 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
2422, 23oveq12d 6622 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
2524, 13syl6eqr 2673 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
2625fveq2d 6152 . . . . . . . . . 10 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27 1stfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝑇)
2826, 27syl6eqr 2673 . . . . . . . . 9 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
2928oveqd 6621 . . . . . . . 8 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
3029reseq2d 5356 . . . . . . 7 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (1st ↾ (𝑥𝐻𝑦)))
3120, 20, 30mpt2eq123dv 6670 . . . . . 6 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
3221, 31opeq12d 4378 . . . . 5 (((𝑐 = 𝐶𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
337, 19, 32csbied2 3542 . . . 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 16734 . . . 4 1stF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ((Base‘𝑐) × (Base‘𝑑)) / 𝑏⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
35 opex 4893 . . . 4 ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ ∈ V
3633, 34, 35ovmpt2a 6744 . . 3 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 1stF 𝐷) = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
372, 3, 36syl2anc 692 . 2 (𝜑 → (𝐶 1stF 𝐷) = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
381, 37syl5eq 2667 1 (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ⦋csb 3514  ⟨cop 4154   × cxp 5072   ↾ cres 5076  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  1st c1st 7111  Basecbs 15781  Hom chom 15873  Catccat 16246   ×c cxpc 16729   1stF c1stf 16730 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-hom 15887  df-cco 15888  df-xpc 16733  df-1stf 16734 This theorem is referenced by:  1stf1  16753  1stf2  16754  1stfcl  16758
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