Theorem cnvf1o 6283

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 2196	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥})
2		snexg 4217	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ V)
3		cnvexg 5207	. . . 4 ⊢ ({𝑥} ∈ V → ◡{𝑥} ∈ V)
4		uniexg 4474	. . . 4 ⊢ (◡{𝑥} ∈ V → ∪ ◡{𝑥} ∈ V)
5	2, 3, 4	3syl 17	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ ◡{𝑥} ∈ V)
6	5	adantl 277	. 2 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∪ ◡{𝑥} ∈ V)
7		snexg 4217	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → {𝑦} ∈ V)
8		cnvexg 5207	. . . 4 ⊢ ({𝑦} ∈ V → ◡{𝑦} ∈ V)
9		uniexg 4474	. . . 4 ⊢ (◡{𝑦} ∈ V → ∪ ◡{𝑦} ∈ V)
10	7, 8, 9	3syl 17	. . 3 ⊢ (𝑦 ∈ ◡𝐴 → ∪ ◡{𝑦} ∈ V)
11	10	adantl 277	. 2 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ ◡𝐴) → ∪ ◡{𝑦} ∈ V)
12		cnvf1olem 6282	. . 3 ⊢ ((Rel 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))
13		relcnv 5047	. . . . 5 ⊢ Rel ◡𝐴
14		simpr 110	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))
15		cnvf1olem 6282	. . . . 5 ⊢ ((Rel ◡𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
16	13, 14, 15	sylancr 414	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
17		dfrel2 5120	. . . . . . 7 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
18		eleq2 2260	. . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴))
19	17, 18	sylbi 121	. . . . . 6 ⊢ (Rel 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴))
20	19	anbi1d 465	. . . . 5 ⊢ (Rel 𝐴 → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})))
21	20	adantr 276	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})))
22	16, 21	mpbid 147	. . 3 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))
23	12, 22	impbida 596	. 2 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})))
24	1, 6, 11, 23	f1od 6126	1 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  Vcvv 2763  {csn 3622  ∪ cuni 3839   ↦ cmpt 4094  ^◡ccnv 4662  Rel wrel 4668  –1-1-onto→wf1o 5257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199

This theorem is referenced by:  tposf12  6327  cnven  6867  xpcomf1o  6884  fsumcnv  11602  fprodcnv  11790