Theorem swapfval 48941

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 → (𝐶swapF𝐷) = ⟨(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 48939	. . 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 7464	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) ∈ V)
4		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	oveq12d 7447	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
7		swapfval.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
8	6, 7	eqtr4di 2794	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) = 𝑆)
9		fvexd 6919	. . . 4 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) ∈ V)
10		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
11	10	fveq2d 6908	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
12		swapfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	eqtr4di 2794	. . . 4 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) = 𝐵)
14		fvexd 6919	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) ∈ V)
15		simplr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → 𝑠 = 𝑆)
16	15	fveq2d 6908	. . . . . 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 730	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → 𝐻 = (Hom ‘𝑆))
19	16, 18	eqtr4d 2779	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = 𝐻)
20		simplr 769	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑏 = 𝐵)
21	20	mpteq1d 5235	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
22		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻)
23	22	oveqd 7446	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑢ℎ𝑣) = (𝑢𝐻𝑣))
24	23	mpteq1d 5235	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))
25	20, 20, 24	mpoeq123dv 7506	. . . . . 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 4879	. . . . 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 3935	. . . 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 3935	. . 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 3935	. 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 3503	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
32		swapfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
33	32	elexd 3503	. 2 ⊢ (𝜑 → 𝐷 ∈ V)
34		opex 5467	. . 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 7582	1 ⊢ (𝜑 → (𝐶swapF𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3479  ⦋csb 3898  {csn 4624  ⟨cop 4630  ∪ cuni 4905   ↦ cmpt 5223  ^◡ccnv 5682  ‘cfv 6559  (class class class)co 7429   ∈ cmpo 7431  Basecbs 17243  Hom chom 17304   ×_c cxpc 18209  swap_Fcswapf 48938

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6512  df-fun 6561  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-swapf 48939

This theorem is referenced by:  swapfelvv  48942  swapf2fvala  48943  swapf1vala  48945