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Theorem tposf2 7579
Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposf2 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))

Proof of Theorem tposf2
StepHypRef Expression
1 ffn 6223 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dffn4 6304 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
31, 2sylib 209 . . . . . 6 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
4 tposfo2 7578 . . . . . 6 (Rel 𝐴 → (𝐹:𝐴onto→ran 𝐹 → tpos 𝐹:𝐴onto→ran 𝐹))
53, 4syl5 34 . . . . 5 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴onto→ran 𝐹))
65imp 395 . . . 4 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴onto→ran 𝐹)
7 fof 6298 . . . 4 (tpos 𝐹:𝐴onto→ran 𝐹 → tpos 𝐹:𝐴⟶ran 𝐹)
86, 7syl 17 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴⟶ran 𝐹)
9 frn 6229 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
109adantl 473 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → ran 𝐹𝐵)
118, 10fssd 6237 . 2 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴𝐵)
1211ex 401 1 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wss 3732  ccnv 5276  ran crn 5278  Rel wrel 5282   Fn wfn 6063  wf 6064  ontowfo 6066  tpos ctpos 7554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-tpos 7555
This theorem is referenced by:  tposf  7583
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