Theorem swapfelvv 49845

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 49844	. 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 6875	. . . 4 ⊢ (Base‘(𝐶 ×c 𝐷)) ∈ V
8	7	mptex 7202	. . 3 ⊢ (𝑥 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ ∪ ◡{𝑥}) ∈ V
9	7, 7	mpoex 8055	. . 3 ⊢ (𝑢 ∈ (Base‘(𝐶 ×c 𝐷)), 𝑣 ∈ (Base‘(𝐶 ×c 𝐷)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝐶 ×c 𝐷))𝑣) ↦ ∪ ◡{𝑓})) ∈ V
10	8, 9	opelvv 5683	. 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))

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 3453  {csn 4579  ⟨cop 4585  ∪ cuni 4862   ↦ cmpt 5178   × cxp 5641  ^◡ccnv 5642  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  Basecbs 17236  Hom chom 17288   ×_c cxpc 18191   swap_F cswapf 49841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-swapf 49842

This theorem is referenced by:  swapf2fval  49847  swapf1val  49849  swapffunca  49866  swapfiso  49867  cofuswapf1  49876  cofuswapf2  49877