Theorem 1stfval 18152

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 6871	. . . . . . 7 ⊢ (Base‘𝑐) ∈ V
5		fvex 6871	. . . . . . 7 ⊢ (Base‘𝑑) ∈ V
6	4, 5	xpex 7729	. . . . . 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 6862	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		simpr 484	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
11	10	fveq2d 6862	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
12	9, 11	xpeq12d 5669	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13		1stfval.t	. . . . . . . 8 ⊢ 𝑇 = (𝐶 ×c 𝐷)
14		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	13, 14, 15	xpcbas 18139	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17		1stfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑇)
18	16, 17	eqtr4i 2755	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
19	12, 18	eqtrdi 2780	. . . . 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 5950	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ 𝑏) = (1st ↾ 𝐵))
22		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
24	22, 23	oveq12d 7405	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
25	24, 13	eqtr4di 2782	. . . . . . . . . . 11 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
26	25	fveq2d 6862	. . . . . . . . . 10 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27		1stfval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝑇)
28	26, 27	eqtr4di 2782	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
29	28	oveqd 7404	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
30	29	reseq2d 5950	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (1st ↾ (𝑥𝐻𝑦)))
31	20, 20, 30	mpoeq123dv 7464	. . . . . 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 4845	. . . . 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 3899	. . . 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 18134	. . . 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 5424	. . . 4 ⊢ ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ ∈ V
36	33, 34, 35	ovmpoa 7544	. . 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 2776	1 ⊢ (𝜑 → 𝑃 = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⦋csb 3862  ⟨cop 4595   × cxp 5636   ↾ cres 5640  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  Basecbs 17179  Hom chom 17231  Catccat 17625   ×_c cxpc 18129   1^st_F c1stf 18130

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-xpc 18133  df-1stf 18134

This theorem is referenced by:  1stf1  18153  1stf2  18154  1stfcl  18158  oppc1stf  49277  1stfpropd  49279