Theorem dfswapf2 49620

Description: Alternate definition of swap_F (df-swapf 49619). (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 49619	. 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 6855	. . . . . 6 ⊢ (Base‘(𝑐 ×c 𝑑)) ∈ V
3		id 22	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = (Base‘(𝑐 ×c 𝑑)))
4		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑐 ×c 𝑑) = (𝑐 ×c 𝑑)
5		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝑐) = (Base‘𝑐)
6		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝑑) = (Base‘𝑑)
7	4, 5, 6	xpcbas 18113	. . . . . . . . . 10 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = (Base‘(𝑐 ×c 𝑑))
8	3, 7	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = ((Base‘𝑐) × (Base‘𝑑)))
9	8	mpteq1d 5190	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}))
10		eqidd 2738	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))
11	8, 8, 10	mpoeq123dv 7443	. . . . . . . 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 4839	. . . . . . 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 3858	. . . . . 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 3886	. . . . 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 7401	. . . . . 6 ⊢ (𝑐 ×c 𝑑) ∈ V
16		fveq2 6842	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Base‘𝑠) = (Base‘(𝑐 ×c 𝑑)))
17		fveq2 6842	. . . . . . . 8 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Hom ‘𝑠) = (Hom ‘(𝑐 ×c 𝑑)))
18	17	csbeq1d 3855	. . . . . . 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 3860	. . . . . 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 3886	. . . . 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 3855	. . . . . . . 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 3860	. . . . . . 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 3886	. . . . . 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 5946	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ 𝑏) = (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))))
25		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢ℎ𝑣)))
26	8, 8, 25	mpoeq123dv 7443	. . . . . . . . 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 4839	. . . . . . . 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 3858	. . . . . . 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 3886	. . . . . 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 2737	. . . . . . . . 9 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝑐) × (Base‘𝑑))
31	30	tposideq2 49248	. . . . . . . 8 ⊢ (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥})
32		eqid 2737	. . . . . . . . . . 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 49248	. . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑐) = (Hom ‘𝑐)
35		eqid 2737	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑑) = (Hom ‘𝑑)
36		eqid 2737	. . . . . . . . . . . 12 ⊢ (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘(𝑐 ×c 𝑑))
37		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → 𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)))
38		simpr 484	. . . . . . . . . . . 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 18115	. . . . . . . . . . 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 5946	. . . . . . . . . 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 5190	. . . . . . . . . 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 2798	. . . . . . . . 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 7446	. . . . . . . 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 4836	. . . . . . 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 6855	. . . . . . . 8 ⊢ (Hom ‘(𝑐 ×c 𝑑)) ∈ V
46		oveq 7374	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢ℎ𝑣) = (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))
47	46	reseq2d 5946	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))
48	47	mpoeq3dv 7447	. . . . . . . . 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 4838	. . . . . . . 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 3886	. . . . . . 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 5190	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
52	51	mpoeq3dv 7447	. . . . . . . . 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 4838	. . . . . . . 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 3886	. . . . . . 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 2770	. . . . . 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 2764	. . . . 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 2771	. . . 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 7446	. 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 2763	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)))⟩)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⦋csb 3851  {csn 4582  ⟨cop 4588  ∪ cuni 4865   ↦ cmpt 5181   I cid 5526   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942  tpos ctpos 8177  Basecbs 17148  Hom chom 17200   ×_c cxpc 18103   swap_F cswapf 49618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-hom 17213  df-cco 17214  df-xpc 18107  df-swapf 49619

This theorem is referenced by: (None)