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Mirrors > Home > MPE Home > Th. List > Mathboxes > nvocnvb | Structured version Visualization version GIF version |
Description: Equivalence to saying the converse of an involution is the function itself. (Contributed by RP, 13-Oct-2024.) |
Ref | Expression |
---|---|
nvocnvb | ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvof1o 7231 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴) | |
2 | fveq1 6846 | . . . . . 6 ⊢ (◡𝐹 = 𝐹 → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥))) | |
3 | 2 | ad2antlr 726 | . . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑥))) |
4 | f1ocnvfv1 7227 | . . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥) | |
5 | 1, 4 | sylan 581 | . . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝑥)) = 𝑥) |
6 | 3, 5 | eqtr3d 2779 | . . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝐹‘𝑥)) = 𝑥) |
7 | 6 | ralrimiva 3144 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) |
8 | 1, 7 | jca 513 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (𝐹:𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)) |
9 | f1of 6789 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴) | |
10 | ffn 6673 | . . . . 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 7232 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹) | |
13 | 11, 12 | jca 513 | . . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → (𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹)) |
14 | 9, 13 | sylan 581 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ◡ccnv 5637 Fn wfn 6496 ⟶wf 6497 –1-1-onto→wf1o 6500 ‘cfv 6501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 |
This theorem is referenced by: (None) |
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