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Theorem f1omvdmvd 19160
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 486 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ dom (𝐹 ∖ I ))
2 f1ofn 6781 . . . . . 6 (𝐹:𝐴1-1-onto𝐴𝐹 Fn 𝐴)
3 difss 4090 . . . . . . . . 9 (𝐹 ∖ I ) ⊆ 𝐹
4 dmss 5855 . . . . . . . . 9 ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
53, 4ax-mp 5 . . . . . . . 8 dom (𝐹 ∖ I ) ⊆ dom 𝐹
6 f1odm 6784 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴 → dom 𝐹 = 𝐴)
75, 6sseqtrid 3995 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
87sselda 3943 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋𝐴)
9 fnelnfp 7118 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
102, 8, 9syl2an2r 684 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
111, 10mpbid 231 . . . 4 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ≠ 𝑋)
12 f1of1 6779 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴1-1𝐴)
1312adantr 482 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴1-1𝐴)
14 f1of 6780 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
1514adantr 482 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴𝐴)
1615, 8ffvelcdmd 7031 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ 𝐴)
17 f1fveq 7204 . . . . . 6 ((𝐹:𝐴1-1𝐴 ∧ ((𝐹𝑋) ∈ 𝐴𝑋𝐴)) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1813, 16, 8, 17syl12anc 836 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1918necon3bid 2987 . . . 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 7118 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝑋) ∈ 𝐴) → ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋)))
222, 16, 21syl2an2r 684 . . 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 4746 . 2 ((𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}) ↔ ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ∧ (𝐹𝑋) ≠ 𝑋))
2523, 11, 24sylanbrc 584 1 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2942  cdif 3906  wss 3909  {csn 4585   I cid 5528  dom cdm 5631   Fn wfn 6487  wf 6488  1-1wf1 6489  1-1-ontowf1o 6491  cfv 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-f1o 6499  df-fv 6500
This theorem is referenced by:  f1otrspeq  19164  symggen  19187  pmtrcnel  31741  pmtrcnelor  31743
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