Theorem swapfval 50073

Description: Value of the swap functor. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
swapfval.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapfval.b	⊢ 𝐵 = (Base‘𝑆)
swapfval.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))

Assertion

Ref	Expression
swapfval	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)

Distinct variable groups:   𝑢,𝐵,𝑣,𝑥   𝑢,𝐶,𝑣   𝑢,𝐷,𝑣   𝑓,𝐻,𝑢,𝑣   𝑢,𝑆,𝑣   𝜑,𝑢,𝑣   𝑥,𝑓

Allowed substitution hints:   𝜑(𝑥, 𝑓)   𝐵(𝑓)   𝐶(𝑥, 𝑓)   𝐷(𝑥, 𝑓)   𝑆(𝑥, 𝑓)   𝑈(𝑥, 𝑣, 𝑢, 𝑓)   𝐻(𝑥)   𝑉(𝑥, 𝑣, 𝑢, 𝑓)

Proof of Theorem swapfval

Dummy variables 𝑏 𝑐 𝑑 ℎ 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-swapf 50071	. . 3 ⊢ 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	1	a1i 11	. 2 ⊢ (𝜑 → 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}))⟩))
3		ovexd 7451	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) ∈ V)
4		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	oveq12d 7434	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
7		swapfval.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
8	6, 7	eqtr4di 2818	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) = 𝑆)
9		fvexd 6900	. . . 4 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) ∈ V)
10		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
11	10	fveq2d 6889	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
12		swapfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	eqtr4di 2818	. . . 4 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) = 𝐵)
14		fvexd 6900	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) ∈ V)
15		simplr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → 𝑠 = 𝑆)
16	15	fveq2d 6889	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = (Hom ‘𝑆))
17		swapfval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
18	17	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → 𝐻 = (Hom ‘𝑆))
19	16, 18	eqtr4d 2803	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = 𝐻)
20		simplr 781	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑏 = 𝐵)
21	20	mpteq1d 5203	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
22		simpr 490	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻)
23	22	oveqd 7433	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑢ℎ𝑣) = (𝑢𝐻𝑣))
24	23	mpteq1d 5203	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))
25	20, 20, 24	mpoeq123dv 7491	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
26	21, 25	opeq12d 4848	. . . . 5 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
27	14, 19, 26	csbied2 3891	. . . 4 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → ⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
28	9, 13, 27	csbied2 3891	. . 3 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
29	3, 8, 28	csbied2 3891	. 2 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
30		swapfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
31	30	elexd 3480	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
32		swapfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
33	32	elexd 3480	. 2 ⊢ (𝜑 → 𝐷 ∈ V)
34		opex 5447	. . 3 ⊢ ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩ ∈ V
35	34	a1i 11	. 2 ⊢ (𝜑 → ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩ ∈ V)
36	2, 29, 31, 33, 35	ovmpod 7568	1 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⦋csb 3854  {csn 4591  ⟨cop 4597  ∪ cuni 4874   ↦ cmpt 5194  ^◡ccnv 5662  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  Basecbs 17287  Hom chom 17339   ×_c cxpc 18242   swap_F cswapf 50070

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-swapf 50071

This theorem is used by:  swapfelvv  50074  swapf2fvala  50075  swapf1vala  50077