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Theorem tposf2 8274
Description: The domain and codomain 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 tposfo2 8273 . . . . 5 (Rel 𝐴 → (𝐹:𝐴onto→ran 𝐹 → tpos 𝐹:𝐴onto→ran 𝐹))
2 ffn 6737 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 dffn4 6827 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
42, 3sylib 218 . . . . 5 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
51, 4impel 505 . . . 4 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴onto→ran 𝐹)
6 fof 6821 . . . 4 (tpos 𝐹:𝐴onto→ran 𝐹 → tpos 𝐹:𝐴⟶ran 𝐹)
75, 6syl 17 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴⟶ran 𝐹)
8 frn 6744 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
98adantl 481 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → ran 𝐹𝐵)
107, 9fssd 6754 . 2 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴𝐵)
1110ex 412 1 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wss 3963  ccnv 5688  ran crn 5690  Rel wrel 5694   Fn wfn 6558  wf 6559  ontowfo 6561  tpos ctpos 8249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-tpos 8250
This theorem is referenced by:  tposf  8278
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