Theorem f1omvdmvd 19537

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 490	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ dom (𝐹 ∖ I ))
2		f1ofn 6825	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
3		difss 4090	. . . . . . . . 9 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
4		dmss 5894	. . . . . . . . 9 ⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ I ) ⊆ dom 𝐹
6		f1odm 6828	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom 𝐹 = 𝐴)
7	5, 6	sseqtrid 3980	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
8	7	sselda 3938	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ 𝐴)
9		fnelnfp 7179	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑋) ≠ 𝑋))
10	2, 8, 9	syl2an2r 698	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑋) ≠ 𝑋))
11	1, 10	mpbid 235	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ≠ 𝑋)
12		f1of1 6823	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴–1-1→𝐴)
13	12	adantr 486	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴–1-1→𝐴)
14		f1of 6824	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
15	14	adantr 486	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴⟶𝐴)
16	15, 8	ffvelcdmd 7084	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ 𝐴)
17		f1fveq 7262	. . . . . 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 850	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹‘𝑋)) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = 𝑋))
19	18	necon3bid 3004	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹‘𝑋)) ≠ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≠ 𝑋))
20	11, 19	mpbird 260	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹‘𝑋)) ≠ (𝐹‘𝑋))
21		fnelnfp 7179	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹‘𝑋) ∈ 𝐴) → ((𝐹‘𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹‘𝑋)) ≠ (𝐹‘𝑋)))
22	2, 16, 21	syl2an2r 698	. . 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 260	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ dom (𝐹 ∖ I ))
24		eldifsn 4755	. 2 ⊢ ((𝐹‘𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}) ↔ ((𝐹‘𝑋) ∈ dom (𝐹 ∖ I ) ∧ (𝐹‘𝑋) ≠ 𝑋))
25	23, 11, 24	sylanbrc 595	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∖ cdif 3903   ⊆ wss 3906  {csn 4591   I cid 5557  dom cdm 5663   Fn wfn 6535  ⟶wf 6536  –1-1→wf1 6537  –1-1-onto→wf1o 6539  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-f1o 6547  df-fv 6548

This theorem is used by:  f1otrspeq  19541  symggen  19564  pmtrcnel  33449  pmtrcnelor  33451