MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1omvdmvd Structured version   Visualization version   GIF version

Theorem f1omvdmvd 18563
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 488 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ dom (𝐹 ∖ I ))
2 f1ofn 6591 . . . . . 6 (𝐹:𝐴1-1-onto𝐴𝐹 Fn 𝐴)
3 difss 4059 . . . . . . . . 9 (𝐹 ∖ I ) ⊆ 𝐹
4 dmss 5735 . . . . . . . . 9 ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
53, 4ax-mp 5 . . . . . . . 8 dom (𝐹 ∖ I ) ⊆ dom 𝐹
6 f1odm 6594 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴 → dom 𝐹 = 𝐴)
75, 6sseqtrid 3967 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
87sselda 3915 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋𝐴)
9 fnelnfp 6916 . . . . . 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 235 . . . 4 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ≠ 𝑋)
12 f1of1 6589 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴1-1𝐴)
1312adantr 484 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴1-1𝐴)
14 f1of 6590 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
1514adantr 484 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴𝐴)
1615, 8ffvelrnd 6829 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ 𝐴)
17 f1fveq 6998 . . . . . 6 ((𝐹:𝐴1-1𝐴 ∧ ((𝐹𝑋) ∈ 𝐴𝑋𝐴)) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1813, 16, 8, 17syl12anc 835 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1918necon3bid 3031 . . . 4 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋) ↔ (𝐹𝑋) ≠ 𝑋))
2011, 19mpbird 260 . . 3 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋))
21 fnelnfp 6916 . . . 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 260 . 2 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ dom (𝐹 ∖ I ))
24 eldifsn 4680 . 2 ((𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}) ↔ ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ∧ (𝐹𝑋) ≠ 𝑋))
2523, 11, 24sylanbrc 586 1 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wne 2987  cdif 3878  wss 3881  {csn 4525   I cid 5424  dom cdm 5519   Fn wfn 6319  wf 6320  1-1wf1 6321  1-1-ontowf1o 6323  cfv 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-f1o 6331  df-fv 6332
This theorem is referenced by:  f1otrspeq  18567  symggen  18590  pmtrcnel  30783  pmtrcnelor  30785
  Copyright terms: Public domain W3C validator