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Theorem tposfo2 5937
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 5936 . . . 4 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
21adantrd 273 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn 𝐴))
3 fndm 5049 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43releqd 4470 . . . . . . . 8 (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
54biimparc 293 . . . . . . 7 ((Rel 𝐴𝐹 Fn 𝐴) → Rel dom 𝐹)
6 rntpos 5927 . . . . . . 7 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
75, 6syl 14 . . . . . 6 ((Rel 𝐴𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹)
87eqeq1d 2091 . . . . 5 ((Rel 𝐴𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵))
98biimprd 156 . . . 4 ((Rel 𝐴𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵))
109expimpd 355 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵))
112, 10jcad 301 . 2 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵)))
12 df-fo 4958 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
13 df-fo 4958 . 2 (tpos 𝐹:𝐴onto𝐵 ↔ (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵))
1411, 12, 133imtr4g 203 1 (Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  ccnv 4390  dom cdm 4391  ran crn 4392  Rel wrel 4396   Fn wfn 4947  ontowfo 4950  tpos ctpos 5914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-fo 4958  df-fv 4960  df-tpos 5915
This theorem is referenced by:  tposf2  5938  tposf1o2  5940  tposfo  5941
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