Theorem 1stfval 18148

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 6847	. . . . . . 7 ⊢ (Base‘𝑐) ∈ V
5		fvex 6847	. . . . . . 7 ⊢ (Base‘𝑑) ∈ V
6	4, 5	xpex 7700	. . . . . 6 ⊢ ((Base‘𝑐) × (Base‘𝑑)) ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8		simpl 482	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
9	8	fveq2d 6838	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		simpr 484	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
11	10	fveq2d 6838	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
12	9, 11	xpeq12d 5655	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 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)
16	13, 14, 15	xpcbas 18135	. . . . . . 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 484	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
21	20	reseq2d 5938	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ 𝑏) = (1st ↾ 𝐵))
22		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
24	22, 23	oveq12d 7378	. . . . . . . . . . . 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 6838	. . . . . . . . . 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 7377	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
30	29	reseq2d 5938	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (1st ↾ (𝑥𝐻𝑦)))
31	20, 20, 30	mpoeq123dv 7435	. . . . . 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 4825	. . . . 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 3875	. . . 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 18130	. . . 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 5411	. . . 4 ⊢ ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ ∈ V
36	33, 34, 35	ovmpoa 7515	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 1stF 𝐷) = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
37	2, 3, 36	syl2anc 585	. 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 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⦋csb 3838  ⟨cop 4574   × cxp 5622   ↾ cres 5626  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  1^st c1st 7933  Basecbs 17170  Hom chom 17222  Catccat 17621   ×_c cxpc 18125   1^st_F c1stf 18126

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-xpc 18129  df-1stf 18130

This theorem is referenced by:  1stf1  18149  1stf2  18150  1stfcl  18154  oppc1stf  49775  1stfpropd  49777