Theorem nvocnvb 44423

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 7288	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)
2		fveq1 6884	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
3	2	ad2antlr 740	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥)))
4		f1ocnvfv1 7284	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
5	1, 4	sylan 592	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥)
6	3, 5	eqtr3d 2797	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
7	6	ralrimiva 3154	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)
8	1, 7	jca 521	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))
9		f1of 6824	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
10		ffn 6709	. . . . 5 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
11	10	adantr 486	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 Fn 𝐴)
12		nvocnv 7289	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)
13	11, 12	jca 521	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹))
14	9, 13	sylan 592	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹))
15	8, 14	impbii 212	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ^◡ccnv 5649   Fn wfn 6533  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is used by: (None)