Theorem nvocnvb 42731

Description: Equivalence to saying the converse of an involution is the function itself. (Contributed by RP, 13-Oct-2024.)

Assertion

Ref	Expression
nvocnvb	⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem nvocnvb

Step	Hyp	Ref	Expression
1		nvof1o 7273	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)
2		fveq1 6883	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
3	2	ad2antlr 724	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
4		f1ocnvfv1 7269	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
5	1, 4	sylan 579	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
6	3, 5	eqtr3d 2768	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
7	6	ralrimiva 3140	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)
8	1, 7	jca 511	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))
9		f1of 6826	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
10		ffn 6710	. . . . 5 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
11	10	adantr 480	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 Fn 𝐴)
12		nvocnv 7274	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)
13	11, 12	jca 511	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹))
14	9, 13	sylan 579	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹))
15	8, 14	impbii 208	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ^◡ccnv 5668   Fn wfn 6531  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by: (None)