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Theorem tposf2 8257
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 8256 . . . . 5 (Rel 𝐴 → (𝐹:𝐴onto→ran 𝐹 → tpos 𝐹:𝐴onto→ran 𝐹))
2 ffn 6720 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 dffn4 6813 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
42, 3sylib 217 . . . . 5 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
51, 4impel 504 . . . 4 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴onto→ran 𝐹)
6 fof 6807 . . . 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 6727 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
98adantl 480 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → ran 𝐹𝐵)
107, 9fssd 6737 . 2 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴𝐵)
1110ex 411 1 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wss 3946  ccnv 5673  ran crn 5675  Rel wrel 5679   Fn wfn 6541  wf 6542  ontowfo 6544  tpos ctpos 8232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fo 6552  df-fv 6554  df-tpos 8233
This theorem is referenced by:  tposf  8261
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