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Theorem f1omvdmvd 19476
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 484 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ dom (𝐹 ∖ I ))
2 f1ofn 6850 . . . . . 6 (𝐹:𝐴1-1-onto𝐴𝐹 Fn 𝐴)
3 difss 4146 . . . . . . . . 9 (𝐹 ∖ I ) ⊆ 𝐹
4 dmss 5916 . . . . . . . . 9 ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
53, 4ax-mp 5 . . . . . . . 8 dom (𝐹 ∖ I ) ⊆ dom 𝐹
6 f1odm 6853 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴 → dom 𝐹 = 𝐴)
75, 6sseqtrid 4048 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
87sselda 3995 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋𝐴)
9 fnelnfp 7197 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
102, 8, 9syl2an2r 685 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
111, 10mpbid 232 . . . 4 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ≠ 𝑋)
12 f1of1 6848 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴1-1𝐴)
1312adantr 480 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴1-1𝐴)
14 f1of 6849 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
1514adantr 480 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴𝐴)
1615, 8ffvelcdmd 7105 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ 𝐴)
17 f1fveq 7282 . . . . . 6 ((𝐹:𝐴1-1𝐴 ∧ ((𝐹𝑋) ∈ 𝐴𝑋𝐴)) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1813, 16, 8, 17syl12anc 837 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1918necon3bid 2983 . . . 4 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋) ↔ (𝐹𝑋) ≠ 𝑋))
2011, 19mpbird 257 . . 3 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋))
21 fnelnfp 7197 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝑋) ∈ 𝐴) → ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋)))
222, 16, 21syl2an2r 685 . . 3 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋)))
2320, 22mpbird 257 . 2 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ dom (𝐹 ∖ I ))
24 eldifsn 4791 . 2 ((𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}) ↔ ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ∧ (𝐹𝑋) ≠ 𝑋))
2523, 11, 24sylanbrc 583 1 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  cdif 3960  wss 3963  {csn 4631   I cid 5582  dom cdm 5689   Fn wfn 6558  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-f1o 6570  df-fv 6571
This theorem is referenced by:  f1otrspeq  19480  symggen  19503  pmtrcnel  33092  pmtrcnelor  33094
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