Theorem curf1 17477

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 17476	. . 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 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
12	11	oveq1d 7173	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑦))
13	12	mpteq2dv 5164	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)))
14		simp1r 1194	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑥 = 𝑋)
15	14	opeq1d 4811	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
16	14	opeq1d 4811	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
17	15, 16	oveq12d 7176	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩) = (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩))
18	14	fveq2d 6676	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
19		eqidd 2824	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑔 = 𝑔)
20	17, 18, 19	oveq123d 7179	. . . . . 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 5164	. . . . 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	mpoeq3dva 7233	. . . 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 4813	. . 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 5358	. . . 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 6777	. 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 2870	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 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ⟨cop 4575   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  1^st c1st 7689  2^nd c2nd 7690  Basecbs 16485  Hom chom 16578  Catccat 16937  Idccid 16938   Func cfunc 17126   ×_c cxpc 17420   curry_F ccurf 17462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-curf 17466

This theorem is referenced by:  curf11  17478  curf12  17479  curf1cl  17480  curf2ndf  17499