Theorem swapf2fn 50066

Description: The morphism part of the swap functor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
swapf2fvala.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf2fvala.b	⊢ 𝐵 = (Base‘𝑆)
swapf1val.o	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)

Assertion

Ref	Expression
swapf2fn	⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵))

Proof of Theorem swapf2fn

Dummy variables 𝑢 𝑣 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))
2		ovex 7443	. . . 4 ⊢ (𝑢(Hom ‘𝑆)𝑣) ∈ V
3	2	mptex 7221	. . 3 ⊢ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}) ∈ V
4	1, 3	fnmpoi 8063	. 2 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) Fn (𝐵 × 𝐵)
5		swapfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6		swapfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
7		swapf2fvala.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
8		swapf2fvala.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
9		eqidd 2764	. . . 4 ⊢ (𝜑 → (Hom ‘𝑆) = (Hom ‘𝑆))
10		swapf1val.o	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
11	5, 6, 7, 8, 9, 10	swapf2fval 50063	. . 3 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})))
12	11	fneq1d 6628	. 2 ⊢ (𝜑 → (𝑃 Fn (𝐵 × 𝐵) ↔ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) Fn (𝐵 × 𝐵)))
13	4, 12	mpbiri 261	1 ⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {csn 4589  ⟨cop 4595  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17264  Hom chom 17316   ×_c cxpc 18219   swap_F cswapf 50057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-swapf 50058

This theorem is referenced by:  swapffunc  50080