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Theorem tposf2 7528
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 6185 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dffn4 6262 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
31, 2sylib 208 . . . . . 6 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
4 tposfo2 7527 . . . . . 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 393 . . . 4 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴onto→ran 𝐹)
7 fof 6256 . . . 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 6193 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
109adantl 467 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → ran 𝐹𝐵)
118, 10fssd 6197 . 2 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴𝐵)
1211ex 397 1 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wss 3723  ccnv 5248  ran crn 5250  Rel wrel 5254   Fn wfn 6026  wf 6027  ontowfo 6029  tpos ctpos 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-tpos 7504
This theorem is referenced by:  tposf  7532
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