Theorem 2ndfval 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)
2ndfval.p	⊢ 𝑄 = (𝐶 2ndF 𝐷)

Assertion

Ref	Expression
2ndfval	⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (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.

Step	Hyp	Ref	Expression
1		2ndfval.p	. 2 ⊢ 𝑄 = (𝐶 2ndF 𝐷)
2		1stfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		1stfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
4		fvex 6841	. . . . . . 7 ⊢ (Base‘𝑐) ∈ V
5		fvex 6841	. . . . . . 7 ⊢ (Base‘𝑑) ∈ V
6	4, 5	xpex 7697	. . . . . 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 6832	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		simpr 485	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
11	10	fveq2d 6832	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
12	9, 11	xpeq12d 5650	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13		1stfval.t	. . . . . . . 8 ⊢ 𝑇 = (𝐶 ×c 𝐷)
14		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	13, 14, 15	xpcbas 18136	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17		1stfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑇)
18	16, 17	eqtr4i 2765	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
19	12, 18	eqtrdi 2790	. . . . 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 5932	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (2nd ↾ 𝑏) = (2nd ↾ 𝐵))
22		simpll 772	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23		simplr 774	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
24	22, 23	oveq12d 7375	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
25	24, 13	eqtr4di 2792	. . . . . . . . . . 11 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
26	25	fveq2d 6832	. . . . . . . . . 10 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27		1stfval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝑇)
28	26, 27	eqtr4di 2792	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
29	28	oveqd 7374	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
30	29	reseq2d 5932	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (2nd ↾ (𝑥𝐻𝑦)))
31	20, 20, 30	mpoeq123dv 7432	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
32	21, 31	opeq12d 4813	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
33	7, 19, 32	csbied2 3868	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩ = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
34		df-2ndf 18132	. . . 4 ⊢ 2ndF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
35		opex 5404	. . . 4 ⊢ ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ ∈ V
36	33, 34, 35	ovmpoa 7512	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 2ndF 𝐷) = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
37	2, 3, 36	syl2anc 590	. 2 ⊢ (𝜑 → (𝐶 2ndF 𝐷) = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
38	1, 37	eqtrid 2786	1 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⦋csb 3831  ⟨cop 4562   × cxp 5617   ↾ cres 5621  ‘cfv 6486  (class class class)co 7357   ∈ cmpo 7359  2^nd c2nd 7931  Basecbs 17171  Hom chom 17223  Catccat 17622   ×_c cxpc 18126   2^nd_F c2ndf 18128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-slot 17144  df-ndx 17156  df-base 17172  df-hom 17236  df-cco 17237  df-xpc 18130  df-2ndf 18132

This theorem is referenced by:  2ndf1  18153  2ndf2  18154  2ndfcl  18156  oppc2ndf  49787  2ndfpropd  49789