Theorem f1omvdmvd 19424

Description: A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
f1omvdmvd	⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))

Proof of Theorem f1omvdmvd

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ dom (𝐹 ∖ I ))
2		f1ofn 6819	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
3		difss 4111	. . . . . . . . 9 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
4		dmss 5882	. . . . . . . . 9 ⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ I ) ⊆ dom 𝐹
6		f1odm 6822	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom 𝐹 = 𝐴)
7	5, 6	sseqtrid 4001	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
8	7	sselda 3958	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ 𝐴)
9		fnelnfp 7169	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑋) ≠ 𝑋))
10	2, 8, 9	syl2an2r 685	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑋) ≠ 𝑋))
11	1, 10	mpbid 232	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ≠ 𝑋)
12		f1of1 6817	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴–1-1→𝐴)
13	12	adantr 480	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴–1-1→𝐴)
14		f1of 6818	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴⟶𝐴)
16	15, 8	ffvelcdmd 7075	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ 𝐴)
17		f1fveq 7255	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ ((𝐹‘𝑋) ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘(𝐹‘𝑋)) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = 𝑋))
18	13, 16, 8, 17	syl12anc 836	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹‘𝑋)) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = 𝑋))
19	18	necon3bid 2976	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹‘𝑋)) ≠ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≠ 𝑋))
20	11, 19	mpbird 257	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑋)) ≠ (𝐹‘𝑋))
21		fnelnfp 7169	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹‘𝑋) ∈ 𝐴) → ((𝐹‘𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹‘𝑋)) ≠ (𝐹‘𝑋)))
22	2, 16, 21	syl2an2r 685	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹‘𝑋)) ≠ (𝐹‘𝑋)))
23	20, 22	mpbird 257	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ dom (𝐹 ∖ I ))
24		eldifsn 4762	. 2 ⊢ ((𝐹‘𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}) ↔ ((𝐹‘𝑋) ∈ dom (𝐹 ∖ I ) ∧ (𝐹‘𝑋) ≠ 𝑋))
25	23, 11, 24	sylanbrc 583	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   ∖ cdif 3923   ⊆ wss 3926  {csn 4601   I cid 5547  dom cdm 5654   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528  –1-1-onto→wf1o 6530  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-f1o 6538  df-fv 6539

This theorem is referenced by:  f1otrspeq  19428  symggen  19451  pmtrcnel  33100  pmtrcnelor  33102