ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tposfo2 GIF version

Theorem tposfo2 6164
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 6163 . . . 4 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
21adantrd 277 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn 𝐴))
3 fndm 5222 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43releqd 4623 . . . . . . . 8 (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
54biimparc 297 . . . . . . 7 ((Rel 𝐴𝐹 Fn 𝐴) → Rel dom 𝐹)
6 rntpos 6154 . . . . . . 7 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
75, 6syl 14 . . . . . 6 ((Rel 𝐴𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹)
87eqeq1d 2148 . . . . 5 ((Rel 𝐴𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵))
98biimprd 157 . . . 4 ((Rel 𝐴𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵))
109expimpd 360 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵))
112, 10jcad 305 . 2 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵)))
12 df-fo 5129 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
13 df-fo 5129 . 2 (tpos 𝐹:𝐴onto𝐵 ↔ (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵))
1411, 12, 133imtr4g 204 1 (Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  ccnv 4538  dom cdm 4539  ran crn 4540  Rel wrel 4544   Fn wfn 5118  ontowfo 5121  tpos ctpos 6141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fo 5129  df-fv 5131  df-tpos 6142
This theorem is referenced by:  tposf2  6165  tposf1o2  6167  tposfo  6168
  Copyright terms: Public domain W3C validator