Theorem curf1 17251

Description: Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a	⊢ 𝐴 = (Base‘𝐶)
curfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curfval.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b	⊢ 𝐵 = (Base‘𝐷)
curf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
curf1.k	⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
curf1.j	⊢ 𝐽 = (Hom ‘𝐷)
curf1.1	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
curf1	⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)

Distinct variable groups:   𝑦,𝑔,𝑧, 1   𝑦,𝐴   𝐵,𝑔,𝑦,𝑧   𝐶,𝑔,𝑦,𝑧   𝐷,𝑔,𝑦,𝑧   𝜑,𝑔,𝑦,𝑧   𝑔,𝐸,𝑦,𝑧   𝑔,𝐽   𝑔,𝐾,𝑦,𝑧   𝑔,𝑋,𝑦,𝑧   𝑔,𝐹,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑧,𝑔)   𝐺(𝑦,𝑧,𝑔)   𝐽(𝑦,𝑧)

Proof of Theorem curf1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf1.k	. 2 ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
2		curfval.g	. . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
3		curfval.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
4		curfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		curfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		curfval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
7		curfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
8		curf1.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
9		curf1.1	. . . 4 ⊢ 1 = (Id‘𝐶)
10	2, 3, 4, 5, 6, 7, 8, 9	curf1fval 17250	. . 3 ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
11		simpr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
12	11	oveq1d 6937	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑦))
13	12	mpteq2dv 4980	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)))
14		simp1r 1212	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑥 = 𝑋)
15	14	opeq1d 4642	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
16	14	opeq1d 4642	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
17	15, 16	oveq12d 6940	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩) = (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩))
18	14	fveq2d 6450	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
19		eqidd 2778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑔 = 𝑔)
20	17, 18, 19	oveq123d 6943	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔) = (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))
21	20	mpteq2dv 4980	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))
22	21	mpt2eq3dva 6996	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
23	13, 22	opeq12d 4644	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
24		curf1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25		opex 5164	. . . 4 ⊢ ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ ∈ V
26	25	a1i 11	. . 3 ⊢ (𝜑 → ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ ∈ V)
27	10, 23, 24, 26	fvmptd 6548	. 2 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
28	1, 27	syl5eq 2825	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  Vcvv 3397  ⟨cop 4403   ↦ cmpt 4965  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  1^st c1st 7443  2^nd c2nd 7444  Basecbs 16255  Hom chom 16349  Catccat 16710  Idccid 16711   Func cfunc 16899   ×_c cxpc 17194   curry_F ccurf 17236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-curf 17240

This theorem is referenced by:  curf11  17252  curf12  17253  curf1cl  17254  curf2ndf  17273