Theorem 1stfval 17039

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 6342	. . . . . . 7 ⊢ (Base‘𝑐) ∈ V
5		fvex 6342	. . . . . . 7 ⊢ (Base‘𝑑) ∈ V
6	4, 5	xpex 7109	. . . . . 6 ⊢ ((Base‘𝑐) × (Base‘𝑑)) ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) ∈ V)
8		simpl 468	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
9	8	fveq2d 6336	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		simpr 471	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
11	10	fveq2d 6336	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
12	9, 11	xpeq12d 5280	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝐶) × (Base‘𝐷)))
13		1stfval.t	. . . . . . . 8 ⊢ 𝑇 = (𝐶 ×c 𝐷)
14		eqid 2771	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2771	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	13, 14, 15	xpcbas 17026	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
17		1stfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑇)
18	16, 17	eqtr4i 2796	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = 𝐵
19	12, 18	syl6eq 2821	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((Base‘𝑐) × (Base‘𝑑)) = 𝐵)
20		simpr 471	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
21	20	reseq2d 5534	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ 𝑏) = (1st ↾ 𝐵))
22		simpll 750	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
23		simplr 752	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
24	22, 23	oveq12d 6811	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
25	24, 13	syl6eqr 2823	. . . . . . . . . . 11 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑐 ×c 𝑑) = 𝑇)
26	25	fveq2d 6336	. . . . . . . . . 10 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘𝑇))
27		1stfval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝑇)
28	26, 27	syl6eqr 2823	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Hom ‘(𝑐 ×c 𝑑)) = 𝐻)
29	28	oveqd 6810	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦) = (𝑥𝐻𝑦))
30	29	reseq2d 5534	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (1st ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)) = (1st ↾ (𝑥𝐻𝑦)))
31	20, 20, 30	mpt2eq123dv 6864	. . . . . 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 4547	. . . . 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 3710	. . . 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 17021	. . . 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 5060	. . . 4 ⊢ ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ ∈ V
36	33, 34, 35	ovmpt2a 6938	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 1stF 𝐷) = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
37	2, 3, 36	syl2anc 573	. 2 ⊢ (𝜑 → (𝐶 1stF 𝐷) = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
38	1, 37	syl5eq 2817	1 ⊢ (𝜑 → 𝑃 = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Vcvv 3351  ⦋csb 3682  ⟨cop 4322   × cxp 5247   ↾ cres 5251  ‘cfv 6031  (class class class)co 6793   ↦ cmpt2 6795  1^st c1st 7313  Basecbs 16064  Hom chom 16160  Catccat 16532   ×_c cxpc 17016   1^st_F c1stf 17017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-hom 16174  df-cco 16175  df-xpc 17020  df-1stf 17021

This theorem is referenced by:  1stf1  17040  1stf2  17041  1stfcl  17045