Theorem 1stfval 18147

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	⊢ (𝜑 → 𝐶 ∈ Cat)
1stfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
1stfval.p	⊢ 𝑃 = (𝐶 1stF 𝐷)

Assertion

Ref	Expression
1stfval	⊢ (𝜑 → 𝑃 = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (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.

Step	Hyp	Ref	Expression
1		1stfval.p	. 2 ⊢ 𝑃 = (𝐶 1stF 𝐷)
2		1stfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		1stfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
4		fvex 6904	. . . . . . 7 ⊢ (Base‘𝑐) ∈ V
5		fvex 6904	. . . . . . 7 ⊢ (Base‘𝑑) ∈ V
6	4, 5	xpex 7742	. . . . . 6 ⊢ ((Base‘𝑐) × (Base‘𝑑)) ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8		simpl 483	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
9	8	fveq2d 6895	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		simpr 485	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
11	10	fveq2d 6895	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
12	9, 11	xpeq12d 5707	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13		1stfval.t	. . . . . . . 8 ⊢ 𝑇 = (𝐶 ×c 𝐷)
14		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	13, 14, 15	xpcbas 18134	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17		1stfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑇)
18	16, 17	eqtr4i 2763	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
19	12, 18	eqtrdi 2788	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = 𝐵)
20		simpr 485	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
21	20	reseq2d 5981	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ 𝑏) = (1st ↾ 𝐵))
22		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
24	22, 23	oveq12d 7429	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
25	24, 13	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
26	25	fveq2d 6895	. . . . . . . . . 10 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27		1stfval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝑇)
28	26, 27	eqtr4di 2790	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
29	28	oveqd 7428	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
30	29	reseq2d 5981	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (1st ↾ (𝑥𝐻𝑦)))
31	20, 20, 30	mpoeq123dv 7486	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
32	21, 31	opeq12d 4881	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(1st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
33	7, 19, 32	csbied2 3933	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(1st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
34		df-1stf 18129	. . . 4 ⊢ 1stF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(1st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
35		opex 5464	. . . 4 ⊢ ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ ∈ V
36	33, 34, 35	ovmpoa 7565	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 1stF 𝐷) = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
37	2, 3, 36	syl2anc 584	. 2 ⊢ (𝜑 → (𝐶 1stF 𝐷) = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
38	1, 37	eqtrid 2784	1 ⊢ (𝜑 → 𝑃 = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⦋csb 3893  ⟨cop 4634   × cxp 5674   ↾ cres 5678  ‘cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7975  Basecbs 17148  Hom chom 17212  Catccat 17612   ×_c cxpc 18124   1^st_F c1stf 18125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-slot 17119  df-ndx 17131  df-base 17149  df-hom 17225  df-cco 17226  df-xpc 18128  df-1stf 18129

This theorem is referenced by:  1stf1  18148  1stf2  18149  1stfcl  18153