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Theorem f1omvdmvd 18084
 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:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))

Proof of Theorem f1omvdmvd
StepHypRef Expression
1 simpr 479 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ dom (𝐹 ∖ I ))
2 f1ofn 6301 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴𝐹 Fn 𝐴)
32adantr 472 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹 Fn 𝐴)
4 difss 3881 . . . . . . . . 9 (𝐹 ∖ I ) ⊆ 𝐹
5 dmss 5479 . . . . . . . . 9 ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
64, 5ax-mp 5 . . . . . . . 8 dom (𝐹 ∖ I ) ⊆ dom 𝐹
7 f1odm 6304 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴 → dom 𝐹 = 𝐴)
86, 7syl5sseq 3795 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
98sselda 3745 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋𝐴)
10 fnelnfp 6609 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
113, 9, 10syl2anc 696 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
121, 11mpbid 222 . . . 4 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ≠ 𝑋)
13 f1of1 6299 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴1-1𝐴)
1413adantr 472 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴1-1𝐴)
15 f1of 6300 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
1615adantr 472 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴𝐴)
1716, 9ffvelrnd 6525 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ 𝐴)
18 f1fveq 6684 . . . . . 6 ((𝐹:𝐴1-1𝐴 ∧ ((𝐹𝑋) ∈ 𝐴𝑋𝐴)) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1914, 17, 9, 18syl12anc 1475 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
2019necon3bid 2977 . . . 4 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋) ↔ (𝐹𝑋) ≠ 𝑋))
2112, 20mpbird 247 . . 3 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋))
22 fnelnfp 6609 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝑋) ∈ 𝐴) → ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋)))
233, 17, 22syl2anc 696 . . 3 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋)))
2421, 23mpbird 247 . 2 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ dom (𝐹 ∖ I ))
25 eldifsn 4463 . 2 ((𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}) ↔ ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ∧ (𝐹𝑋) ≠ 𝑋))
2624, 12, 25sylanbrc 701 1 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2140   ≠ wne 2933   ∖ cdif 3713   ⊆ wss 3716  {csn 4322   I cid 5174  dom cdm 5267   Fn wfn 6045  ⟶wf 6046  –1-1→wf1 6047  –1-1-onto→wf1o 6049  ‘cfv 6050 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-f1o 6057  df-fv 6058 This theorem is referenced by:  f1otrspeq  18088  symggen  18111
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