Theorem tposcurf1 49261

Description: Value of the object part of the transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.)

Hypotheses

Ref	Expression
tposcurf1.g	⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
tposcurf1.a	⊢ 𝐴 = (Base‘𝐶)
tposcurf1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
tposcurf1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
tposcurf1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
tposcurf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
tposcurf1.k	⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))
tposcurf1.b	⊢ 𝐵 = (Base‘𝐷)
tposcurf1.j	⊢ 𝐽 = (Hom ‘𝐷)
tposcurf1.1	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
tposcurf1	⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)

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

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

Proof of Theorem tposcurf1

Step	Hyp	Ref	Expression
1		tposcurf1.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))
2		tposcurf1.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
3	2	fveq2d 6844	. . . 4 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))))
4	3	fveq1d 6842	. . 3 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))
5		eqid 2729	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))) = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))
6		tposcurf1.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
7		tposcurf1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		tposcurf1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
9		tposcurf1.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
10		eqidd 2730	. . . . 5 ⊢ (𝜑 → (𝐹 ∘func (𝐶 swapF 𝐷)) = (𝐹 ∘func (𝐶 swapF 𝐷)))
11	7, 8, 9, 10	cofuswapfcl 49255	. . . 4 ⊢ (𝜑 → (𝐹 ∘func (𝐶 swapF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
12		tposcurf1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
13		tposcurf1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
14		eqid 2729	. . . 4 ⊢ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋)
15		tposcurf1.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
16		tposcurf1.1	. . . 4 ⊢ 1 = (Id‘𝐶)
17	5, 6, 7, 8, 11, 12, 13, 14, 15, 16	curf1 18162	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋) = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩)
18	1, 4, 17	3eqtrd 2768	. 2 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩)
19	12	fvexi 6854	. . . . . . . . 9 ⊢ 𝐵 ∈ V
20	19	mptex 7179	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)) ∈ V
21	19, 19	mpoex 8037	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))) ∈ V
22	20, 21	op1std 7957	. . . . . . 7 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)))
23	18, 22	syl 17	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)))
24		ovexd 7404	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦) ∈ V)
25	23, 24	fvmpt2d 6963	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦))
26	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
27	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
28	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
29	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
30	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐴)
31	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐾 = ((1st ‘𝐺)‘𝑋))
32		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
33	26, 6, 27, 28, 29, 30, 31, 12, 32	tposcurf11 49259	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑦(1st ‘𝐹)𝑋))
34	25, 33	eqtr3d 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦) = (𝑦(1st ‘𝐹)𝑋))
35	34	mpteq2dva 5195	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)))
36	20, 21	op2ndd 7958	. . . . . . . . . 10 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))))
37	18, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))))
38		ovex 7402	. . . . . . . . . . 11 ⊢ (𝑦𝐽𝑧) ∈ V
39	38	mptex 7179	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) ∈ V
40	39	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) ∈ V)
41	37, 40	ovmpt4d 48826	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(2nd ‘𝐾)𝑧) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))
42		ovexd 7404	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔) ∈ V)
43	41, 42	fvmpt2d 6963	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))
44	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
45	7	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐶 ∈ Cat)
46	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐷 ∈ Cat)
47	9	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
48	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑋 ∈ 𝐴)
49	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐾 = ((1st ‘𝐺)‘𝑋))
50		simplrl 776	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑦 ∈ 𝐵)
51		simplrr 777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑧 ∈ 𝐵)
52		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦𝐽𝑧))
53	44, 6, 45, 46, 47, 48, 49, 12, 50, 15, 16, 51, 52	tposcurf12 49260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))
54	43, 53	eqtr3d 2766	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔) = (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))
55	54	mpteq2dva 5195	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))
56	55	3impb 1114	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))
57	56	mpoeq3dva 7446	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))))
58	35, 57	opeq12d 4841	. 2 ⊢ (𝜑 → ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)
59	18, 58	eqtrd 2764	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  Hom chom 17207  Catccat 17601  Idccid 17602   Func cfunc 17792   ∘_func ccofu 17794   ×_c cxpc 18105   curry_F ccurf 18147   swap_F cswapf 49221

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17605  df-cid 17606  df-func 17796  df-cofu 17798  df-xpc 18109  df-curf 18151  df-swapf 49222

This theorem is referenced by:  precofval  49329  precofvalALT  49330