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Theorem tposf2 6262
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 ffn 5360 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dffn4 5439 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
31, 2sylib 122 . . . . . 6 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
4 tposfo2 6261 . . . . . 6 (Rel 𝐴 → (𝐹:𝐴onto→ran 𝐹 → tpos 𝐹:𝐴onto→ran 𝐹))
53, 4syl5 32 . . . . 5 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴onto→ran 𝐹))
65imp 124 . . . 4 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴onto→ran 𝐹)
7 fof 5433 . . . 4 (tpos 𝐹:𝐴onto→ran 𝐹 → tpos 𝐹:𝐴⟶ran 𝐹)
86, 7syl 14 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴⟶ran 𝐹)
9 frn 5369 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
109adantl 277 . . 3 ((Rel 𝐴𝐹:𝐴𝐵) → ran 𝐹𝐵)
11 fss 5372 . . 3 ((tpos 𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹𝐵) → tpos 𝐹:𝐴𝐵)
128, 10, 11syl2anc 411 . 2 ((Rel 𝐴𝐹:𝐴𝐵) → tpos 𝐹:𝐴𝐵)
1312ex 115 1 (Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wss 3129  ccnv 4621  ran crn 4623  Rel wrel 4627   Fn wfn 5206  wf 5207  ontowfo 5209  tpos ctpos 6238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fo 5217  df-fv 5219  df-tpos 6239
This theorem is referenced by:  tposf  6266
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