Theorem nvocnvb 43405

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 7237	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)
2		fveq1 6839	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
3	2	ad2antlr 727	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
4		f1ocnvfv1 7233	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
5	1, 4	sylan 580	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
6	3, 5	eqtr3d 2766	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
7	6	ralrimiva 3125	. . 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 6782	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
10		ffn 6670	. . . . 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 7238	. . . 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 580	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹))
15	8, 14	impbii 209	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ^◡ccnv 5630   Fn wfn 6494  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by: (None)