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Theorem f1omvdmvd 17795
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 477 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋 ∈ dom (𝐹 ∖ I ))
2 f1ofn 6100 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴𝐹 Fn 𝐴)
32adantr 481 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹 Fn 𝐴)
4 difss 3720 . . . . . . . . 9 (𝐹 ∖ I ) ⊆ 𝐹
5 dmss 5288 . . . . . . . . 9 ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
64, 5ax-mp 5 . . . . . . . 8 dom (𝐹 ∖ I ) ⊆ dom 𝐹
7 f1odm 6103 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴 → dom 𝐹 = 𝐴)
86, 7syl5sseq 3637 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
98sselda 3587 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝑋𝐴)
10 fnelnfp 6403 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
113, 9, 10syl2anc 692 . . . . 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 6098 . . . . . . 7 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴1-1𝐴)
1413adantr 481 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴1-1𝐴)
15 f1of 6099 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
1615adantr 481 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → 𝐹:𝐴𝐴)
1716, 9ffvelrnd 6321 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ 𝐴)
18 f1fveq 6479 . . . . . 6 ((𝐹:𝐴1-1𝐴 ∧ ((𝐹𝑋) ∈ 𝐴𝑋𝐴)) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
1914, 17, 9, 18syl12anc 1321 . . . . 5 ((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → ((𝐹‘(𝐹𝑋)) = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑋))
2019necon3bid 2834 . . . 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 6403 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝑋) ∈ 𝐴) → ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(𝐹𝑋)) ≠ (𝐹𝑋)))
233, 17, 22syl2anc 692 . . 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 4292 . 2 ((𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}) ↔ ((𝐹𝑋) ∈ dom (𝐹 ∖ I ) ∧ (𝐹𝑋) ≠ 𝑋))
2624, 12, 25sylanbrc 697 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 384   = wceq 1480  wcel 1987  wne 2790  cdif 3556  wss 3559  {csn 4153   I cid 4989  dom cdm 5079   Fn wfn 5847  wf 5848  1-1wf1 5849  1-1-ontowf1o 5851  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-f1o 5859  df-fv 5860
This theorem is referenced by:  f1otrspeq  17799  symggen  17822
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