Theorem f1omvdcnv 19052

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 7151	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
2	1	3anidm23 1420	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
3	2	bicomd 222	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥))
4	3	necon3bid 2988	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) ≠ 𝑥))
5	4	rabbidva 3413	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → {𝑥 ∈ 𝐴 ∣ (◡𝐹‘𝑥) ≠ 𝑥} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
6		f1ocnv 6728	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐴)
7		f1ofn 6717	. . 3 ⊢ (◡𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹 Fn 𝐴)
8		fndifnfp 7048	. . 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 6717	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
11		fndifnfp 7048	. . 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 2788	1 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068   ∖ cdif 3884   I cid 5488  ^◡ccnv 5588  dom cdm 5589   Fn wfn 6428  –1-1-onto→wf1o 6432  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by:  f1omvdco2  19056  symgsssg  19075  symgfisg  19076