Theorem f1omvdcnv 19390

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 7237	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
2	1	3anidm23 1424	. . . . 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 2977	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) ≠ 𝑥))
5	4	rabbidva 3407	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → {𝑥 ∈ 𝐴 ∣ (◡𝐹‘𝑥) ≠ 𝑥} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
6		f1ocnv 6796	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐴)
7		f1ofn 6785	. . 3 ⊢ (◡𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹 Fn 𝐴)
8		fndifnfp 7134	. . 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 6785	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
11		fndifnfp 7134	. . 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 2782	1 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401   ∖ cdif 3900   I cid 5528  ^◡ccnv 5633  dom cdm 5634   Fn wfn 6497  –1-1-onto→wf1o 6501  ‘cfv 6502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510

This theorem is referenced by:  f1omvdco2  19394  symgsssg  19413  symgfisg  19414