Theorem nvocnvb 43881

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 7228	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)
2		fveq1 6830	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
3	2	ad2antlr 734	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
4		f1ocnvfv1 7224	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
5	1, 4	sylan 587	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
6	3, 5	eqtr3d 2778	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
7	6	ralrimiva 3133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)
8	1, 7	jca 517	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))
9		f1of 6771	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
10		ffn 6659	. . . . 5 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
11	10	adantr 482	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 Fn 𝐴)
12		nvocnv 7229	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)
13	11, 12	jca 517	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹))
14	9, 13	sylan 587	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹))
15	8, 14	impbii 211	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ^◡ccnv 5620   Fn wfn 6484  ⟶wf 6485  –1-1-onto→wf1o 6488  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497

This theorem is referenced by: (None)