Theorem swapfelvv 50340

Description: A swap functor is an ordered pair. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)

Assertion

Ref	Expression
swapfelvv	⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))

Proof of Theorem swapfelvv

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

Step	Hyp	Ref	Expression
1		swapfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
2		swapfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
3		eqid 2761	. . 3 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
4		eqid 2761	. . 3 ⊢ (Base‘(𝐶 ×c 𝐷)) = (Base‘(𝐶 ×c 𝐷))
5		eqidd 2762	. . 3 ⊢ (𝜑 → (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)))
6	1, 2, 3, 4, 5	swapfval 50339	. 2 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ (Base‘(𝐶 ×c 𝐷)), 𝑣 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝐶 ×c 𝐷))𝑣) ↦ ∪ ◡{𝑓}))⟩)
7		fvex 6896	. . . 4 ⊢ (Base‘(𝐶 ×c 𝐷)) ∈ V
8	7	mptex 7227	. . 3 ⊢ (𝑥 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ ∪ ◡{𝑥}) ∈ V
9	7, 7	mpoex 8090	. . 3 ⊢ (𝑢 ∈ (Base‘(𝐶 ×c 𝐷)), 𝑣 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝐶 ×c 𝐷))𝑣) ↦ ∪ ◡{𝑓})) ∈ V
10	8, 9	opelvv 5691	. 2 ⊢ ⟨(𝑥 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ (Base‘(𝐶 ×c 𝐷)), 𝑣 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝐶 ×c 𝐷))𝑣) ↦ ∪ ◡{𝑓}))⟩ ∈ (V × V)
11	6, 10	eqeltrdi 2869	1 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451  {csn 4584  ⟨cop 4590  ∪ cuni 4867   ↦ cmpt 5186   × cxp 5649  ^◡ccnv 5650  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  Basecbs 17380  Hom chom 17432   ×_c cxpc 18335   swap_F cswapf 50336

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-swapf 50337

This theorem is used by:  swapf2fval  50342  swapf1val  50344  swapffunca  50361  swapfiso  50362  cofuswapf1  50371  cofuswapf2  50372