Theorem cnvf1o 8135

Description: Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)

Assertion

Ref	Expression
cnvf1o	⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem cnvf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2735	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥})
2		vsnex 5440	. . . . 5 ⊢ {𝑥} ∈ V
3	2	cnvex 7948	. . . 4 ⊢ ◡{𝑥} ∈ V
4	3	uniex 7760	. . 3 ⊢ ∪ ◡{𝑥} ∈ V
5	4	a1i 11	. 2 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∪ ◡{𝑥} ∈ V)
6		vsnex 5440	. . . . 5 ⊢ {𝑦} ∈ V
7	6	cnvex 7948	. . . 4 ⊢ ◡{𝑦} ∈ V
8	7	uniex 7760	. . 3 ⊢ ∪ ◡{𝑦} ∈ V
9	8	a1i 11	. 2 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ ◡𝐴) → ∪ ◡{𝑦} ∈ V)
10		cnvf1olem 8134	. . 3 ⊢ ((Rel 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))
11		relcnv 6125	. . . . 5 ⊢ Rel ◡𝐴
12		simpr 484	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))
13		cnvf1olem 8134	. . . . 5 ⊢ ((Rel ◡𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
14	11, 12, 13	sylancr 587	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
15		dfrel2 6211	. . . . . . 7 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
16		eleq2 2828	. . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴))
17	15, 16	sylbi 217	. . . . . 6 ⊢ (Rel 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴))
18	17	anbi1d 631	. . . . 5 ⊢ (Rel 𝐴 → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})))
19	18	adantr 480	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})))
20	14, 19	mpbid 232	. . 3 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
21	10, 20	impbida 801	. 2 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})))
22	1, 5, 9, 21	f1od 7685	1 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ∪ cuni 4912   ↦ cmpt 5231  ^◡ccnv 5688  Rel wrel 5694  –1-1-onto→wf1o 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014

This theorem is referenced by:  tposf12  8275  cnven  9072  xpcomf1o  9100  fsumcnv  15806  fprodcnv  16016  gsumcom2  20008