Theorem dfswapf2 50187

Description: Alternate definition of swap_F (df-swapf 50186). (Contributed by Zhi Wang, 9-Oct-2025.)

Assertion

Ref	Expression
dfswapf2	⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)

Distinct variable group:   𝑏,𝑐,𝑑,ℎ,𝑠,𝑢,𝑣

Proof of Theorem dfswapf2

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-swapf 50186	. 2 ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
2		fvex 6891	. . . . . 6 ⊢ (Base‘(𝑐 ×c 𝑑)) ∈ V
3		id 23	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = (Base‘(𝑐 ×c 𝑑)))
4		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑐 ×c 𝑑) = (𝑐 ×c 𝑑)
5		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑐) = (Base‘𝑐)
6		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑑) = (Base‘𝑑)
7	4, 5, 6	xpcbas 18266	. . . . . . . . . 10 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = (Base‘(𝑐 ×c 𝑑))
8	3, 7	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = ((Base‘𝑐) × (Base‘𝑑)))
9	8	mpteq1d 5195	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}))
10		eqidd 2761	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))
11	8, 8, 10	mpoeq123dv 7488	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})))
12	9, 11	opeq12d 4841	. . . . . . 7 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
13	12	csbeq2dv 3854	. . . . . 6 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
14	2, 13	csbie 3882	. . . . 5 ⊢ ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
15		ovex 7446	. . . . . 6 ⊢ (𝑐 ×c 𝑑) ∈ V
16		fveq2 6878	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Base‘𝑠) = (Base‘(𝑐 ×c 𝑑)))
17		fveq2 6878	. . . . . . . 8 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Hom ‘𝑠) = (Hom ‘(𝑐 ×c 𝑑)))
18	17	csbeq1d 3851	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
19	16, 18	csbeq12dv 3856	. . . . . 6 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
20	15, 19	csbie 3882	. . . . 5 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
21	17	csbeq1d 3851	. . . . . . . 8 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
22	16, 21	csbeq12dv 3856	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
23	15, 22	csbie 3882	. . . . . 6 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩
24	8	reseq2d 5972	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ 𝑏) = (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))))
25		eqidd 2761	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢ℎ𝑣)))
26	8, 8, 25	mpoeq123dv 7488	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣))) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣))))
27	24, 26	opeq12d 4841	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
28	27	csbeq2dv 3854	. . . . . . 7 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
29	2, 28	csbie 3882	. . . . . 6 ⊢ ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩
30		eqid 2760	. . . . . . . . 9 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝑐) × (Base‘𝑑))
31	30	tposideq2 49815	. . . . . . . 8 ⊢ (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥})
32		eqid 2760	. . . . . . . . . . 11 ⊢ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))) = (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣)))
33	32	tposideq2 49815	. . . . . . . . . 10 ⊢ (tpos I ↾ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣)))) = (𝑓 ∈ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))) ↦ ∪ ◡{𝑓})
34		eqid 2760	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑐) = (Hom ‘𝑐)
35		eqid 2760	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑑) = (Hom ‘𝑑)
36		eqid 2760	. . . . . . . . . . . 12 ⊢ (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘(𝑐 ×c 𝑑))
37		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → 𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)))
38		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)))
39	4, 7, 34, 35, 36, 37, 38	xpchom 18268	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) = (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))))
40	39	reseq2d 5972	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)) = (tpos I ↾ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣)))))
41	39	mpteq1d 5195	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))) ↦ ∪ ◡{𝑓}))
42	33, 40, 41	3eqtr4a 2821	. . . . . . . . 9 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)) = (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
43	42	mpoeq3ia 7491	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
44	31, 43	opeq12i 4838	. . . . . . 7 ⊢ ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))⟩
45		fvex 6891	. . . . . . . 8 ⊢ (Hom ‘(𝑐 ×c 𝑑)) ∈ V
46		oveq 7419	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢ℎ𝑣) = (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))
47	46	reseq2d 5972	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))
48	47	mpoeq3dv 7492	. . . . . . . . 9 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣))) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))))
49	48	opeq2d 4840	. . . . . . . 8 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))⟩)
50	45, 49	csbie 3882	. . . . . . 7 ⊢ ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))⟩
51	46	mpteq1d 5195	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
52	51	mpoeq3dv 7492	. . . . . . . . 9 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓})))
53	52	opeq2d 4840	. . . . . . . 8 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))⟩)
54	45, 53	csbie 3882	. . . . . . 7 ⊢ ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))⟩
55	44, 50, 54	3eqtr4i 2793	. . . . . 6 ⊢ ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
56	23, 29, 55	3eqtri 2787	. . . . 5 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
57	14, 20, 56	3eqtr4ri 2794	. . . 4 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
58	57	a1i 11	. . 3 ⊢ ((𝑐 ∈ V ∧ 𝑑 ∈ V) → ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
59	58	mpoeq3ia 7491	. 2 ⊢ (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩) = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
60	1, 59	eqtr4i 2786	1 ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⦋csb 3847  {csn 4584  ⟨cop 4590  ∪ cuni 4867   ↦ cmpt 5186   I cid 5549   × cxp 5653  ^◡ccnv 5654   ↾ cres 5657  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7984  2^nd c2nd 7985  tpos ctpos 8223  Basecbs 17301  Hom chom 17353   ×_c cxpc 18256   swap_F cswapf 50185

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 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-struct 17239  df-slot 17274  df-ndx 17286  df-base 17302  df-hom 17366  df-cco 17367  df-xpc 18260  df-swapf 50186

This theorem is used by: (None)