Theorem swapfelvv 49234

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 2730	. . 3 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
4		eqid 2730	. . 3 ⊢ (Base‘(𝐶 ×c 𝐷)) = (Base‘(𝐶 ×c 𝐷))
5		eqidd 2731	. . 3 ⊢ (𝜑 → (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)))
6	1, 2, 3, 4, 5	swapfval 49233	. 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 6873	. . . 4 ⊢ (Base‘(𝐶 ×c 𝐷)) ∈ V
8	7	mptex 7199	. . 3 ⊢ (𝑥 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ ∪ ◡{𝑥}) ∈ V
9	7, 7	mpoex 8060	. . 3 ⊢ (𝑢 ∈ (Base‘(𝐶 ×c 𝐷)), 𝑣 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝐶 ×c 𝐷))𝑣) ↦ ∪ ◡{𝑓})) ∈ V
10	8, 9	opelvv 5680	. 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 2837	1 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3450  {csn 4591  ⟨cop 4597  ∪ cuni 4873   ↦ cmpt 5190   × cxp 5638  ^◡ccnv 5639  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  Basecbs 17185  Hom chom 17237   ×_c cxpc 18135   swap_F cswapf 49230

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-swapf 49231

This theorem is referenced by:  swapf2fval  49236  swapf1val  49238  swapffunca  49255  swapfiso  49256  cofuswapf1  49265  cofuswapf2  49266