Theorem cnvf1o 8112

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 2762	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥})
2		vsnex 5404	. . . . 5 ⊢ {𝑥} ∈ V
3	2	cnvex 7926	. . . 4 ⊢ ◡{𝑥} ∈ V
4	3	uniex 7747	. . 3 ⊢ ∪ ◡{𝑥} ∈ V
5	4	a1i 11	. 2 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∪ ◡{𝑥} ∈ V)
6		vsnex 5404	. . . . 5 ⊢ {𝑦} ∈ V
7	6	cnvex 7926	. . . 4 ⊢ ◡{𝑦} ∈ V
8	7	uniex 7747	. . 3 ⊢ ∪ ◡{𝑦} ∈ V
9	8	a1i 11	. 2 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ ◡𝐴) → ∪ ◡{𝑦} ∈ V)
10		cnvf1olem 8111	. . 3 ⊢ ((Rel 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))
11		relcnv 6104	. . . . 5 ⊢ Rel ◡𝐴
12		simpr 490	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))
13		cnvf1olem 8111	. . . . 5 ⊢ ((Rel ◡𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
14	11, 12, 13	sylancr 599	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
15		dfrel2 6186	. . . . . . 7 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
16		eleq2 2851	. . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴))
17	15, 16	sylbi 220	. . . . . 6 ⊢ (Rel 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴))
18	17	anbi1d 643	. . . . 5 ⊢ (Rel 𝐴 → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})))
19	18	adantr 486	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})))
20	14, 19	mpbid 235	. . 3 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
21	10, 20	impbida 813	. 2 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})))
22	1, 5, 9, 21	f1od 7670	1 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  {csn 4587  ∪ cuni 4870   ↦ cmpt 5190  ^◡ccnv 5658  Rel wrel 5664  –1-1-onto→wf1o 6536

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  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-1st 7990  df-2nd 7991

This theorem is used by:  tposf12  8253  cnven  9044  xpcomf1o  9068  fsumcnv  15863  fprodcnv  16076  gsumcom2  20108  tposres3  49815