Theorem f1omvdcnv 17687

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 6435	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
2	1	3anidm23 1377	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
3	2	bicomd 212	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥))
4	3	necon3bid 2826	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) ≠ 𝑥))
5	4	rabbidva 3163	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → {𝑥 ∈ 𝐴 ∣ (◡𝐹‘𝑥) ≠ 𝑥} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
6		f1ocnv 6062	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐴)
7		f1ofn 6051	. . 3 ⊢ (◡𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹 Fn 𝐴)
8		fndifnfp 6347	. . 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 6051	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
11		fndifnfp 6347	. . 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 2654	1 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537   I cid 4948  ^◡ccnv 5037  dom cdm 5038   Fn wfn 5799  –1-1-onto→wf1o 5803  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812

This theorem is referenced by:  f1omvdco2  17691  symgsssg  17710  symgfisg  17711