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Theorem tposfo2 5882
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfo2 (Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))

Proof of Theorem tposfo2
StepHypRef Expression
1 tposfn2 5881 . . . 4 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
21adantrd 264 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn 𝐴))
3 fndm 4998 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43releqd 4424 . . . . . . . 8 (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
54biimparc 283 . . . . . . 7 ((Rel 𝐴𝐹 Fn 𝐴) → Rel dom 𝐹)
6 rntpos 5872 . . . . . . 7 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
75, 6syl 14 . . . . . 6 ((Rel 𝐴𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹)
87eqeq1d 2048 . . . . 5 ((Rel 𝐴𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵))
98biimprd 147 . . . 4 ((Rel 𝐴𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵))
109expimpd 345 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵))
112, 10jcad 291 . 2 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵)))
12 df-fo 4908 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
13 df-fo 4908 . 2 (tpos 𝐹:𝐴onto𝐵 ↔ (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵))
1411, 12, 133imtr4g 194 1 (Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  ccnv 4344  dom cdm 4345  ran crn 4346  Rel wrel 4350   Fn wfn 4897  ontowfo 4900  tpos ctpos 5859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fo 4908  df-fv 4910  df-tpos 5860
This theorem is referenced by:  tposf2  5883  tposf1o2  5885  tposfo  5886
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