Theorem tposcurf1 50212

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 6886	. . . 4 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))))
4	3	fveq1d 6884	. . 3 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))
5		eqid 2762	. . . 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 2763	. . . . 5 ⊢ (𝜑 → (𝐹 ∘func (𝐶 swapF 𝐷)) = (𝐹 ∘func (𝐶 swapF 𝐷)))
11	7, 8, 9, 10	cofuswapfcl 50206	. . . 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 2762	. . . 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 18317	. . 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 2801	. 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 6896	. . . . . . . . 9 ⊢ 𝐵 ∈ V
20	19	mptex 7225	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)) ∈ V
21	19, 19	mpoex 8081	. . . . . . . 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 7999	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)))
24		ovexd 7451	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦) ∈ V)
25	23, 24	fvmpt2d 7004	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦))
26	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
27	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
28	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
29	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
30	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐴)
31	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐾 = ((1st ‘𝐺)‘𝑋))
32		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
33	26, 6, 27, 28, 29, 30, 31, 12, 32	tposcurf11 50210	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑦(1st ‘𝐹)𝑋))
34	25, 33	eqtr3d 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦) = (𝑦(1st ‘𝐹)𝑋))
35	34	mpteq2dva 5202	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)))
36	20, 21	op2ndd 8000	. . . . . . . . . 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 18	. . . . . . . . 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 7449	. . . . . . . . . . 11 ⊢ (𝑦𝐽𝑧) ∈ V
39	38	mptex 7225	. . . . . . . . . 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 49780	. . . . . . . 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 7451	. . . . . . . 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 7004	. . . . . . 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 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
45	7	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐶 ∈ Cat)
46	8	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐷 ∈ Cat)
47	9	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
48	13	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑋 ∈ 𝐴)
49	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐾 = ((1st ‘𝐺)‘𝑋))
50		simplrl 789	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑦 ∈ 𝐵)
51		simplrr 790	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑧 ∈ 𝐵)
52		simpr 490	. . . . . . . 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 50211	. . . . . . 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 2799	. . . . . 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 5202	. . . . 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 1132	. . . 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 7493	. . 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 4844	. 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 2797	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7987  2^nd c2nd 7988  Basecbs 17305  Hom chom 17357  Catccat 17756  Idccid 17757   Func cfunc 17947   ∘_func ccofu 17949   ×_c cxpc 18260   curry_F ccurf 18302   swap_F cswapf 50172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13564  df-struct 17243  df-slot 17278  df-ndx 17290  df-base 17306  df-hom 17370  df-cco 17371  df-cat 17760  df-cid 17761  df-func 17951  df-cofu 17953  df-xpc 18264  df-curf 18306  df-swapf 50173

This theorem is used by:  precofval  50280  precofvalALT  50281