Theorem f1omvdcnv 19486

Description: A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
f1omvdcnv	⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))

Proof of Theorem f1omvdcnv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ocnvfvb 7315	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
2	1	3anidm23 1421	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
3	2	bicomd 223	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥))
4	3	necon3bid 2991	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) ≠ 𝑥))
5	4	rabbidva 3450	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → {𝑥 ∈ 𝐴 ∣ (◡𝐹‘𝑥) ≠ 𝑥} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
6		f1ocnv 6874	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐴)
7		f1ofn 6863	. . 3 ⊢ (◡𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹 Fn 𝐴)
8		fndifnfp 7210	. . 3 ⊢ (◡𝐹 Fn 𝐴 → dom (◡𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (◡𝐹‘𝑥) ≠ 𝑥})
9	6, 7, 8	3syl 18	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (◡𝐹‘𝑥) ≠ 𝑥})
10		f1ofn 6863	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
11		fndifnfp 7210	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
12	10, 11	syl 17	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
13	5, 9, 12	3eqtr4d 2790	1 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  {crab 3443   ∖ cdif 3973   I cid 5592  ^◡ccnv 5699  dom cdm 5700   Fn wfn 6568  –1-1-onto→wf1o 6572  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  f1omvdco2  19490  symgsssg  19509  symgfisg  19510