Theorem f1omvdcnv 18574

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 7038	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
2	1	3anidm23 1417	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ (◡𝐹‘𝑥) = 𝑥))
3	2	bicomd 225	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥))
4	3	necon3bid 3062	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) ≠ 𝑥))
5	4	rabbidva 3480	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → {𝑥 ∈ 𝐴 ∣ (◡𝐹‘𝑥) ≠ 𝑥} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
6		f1ocnv 6629	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐴)
7		f1ofn 6618	. . 3 ⊢ (◡𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹 Fn 𝐴)
8		fndifnfp 6940	. . 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 6618	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
11		fndifnfp 6940	. . 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 2868	1 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  {crab 3144   ∖ cdif 3935   I cid 5461  ^◡ccnv 5556  dom cdm 5557   Fn wfn 6352  –1-1-onto→wf1o 6356  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365

This theorem is referenced by:  f1omvdco2  18578  symgsssg  18597  symgfisg  18598