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Theorem tposfn2 5911
Description: The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfn2 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))

Proof of Theorem tposfn2
StepHypRef Expression
1 tposfun 5905 . . . 4 (Fun 𝐹 → Fun tpos 𝐹)
21a1i 9 . . 3 (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3 dmtpos 5901 . . . . . 6 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
43a1i 9 . . . . 5 (dom 𝐹 = 𝐴 → (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹))
5 releq 4449 . . . . 5 (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6 cnveq 4536 . . . . . 6 (dom 𝐹 = 𝐴dom 𝐹 = 𝐴)
76eqeq2d 2067 . . . . 5 (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = dom 𝐹 ↔ dom tpos 𝐹 = 𝐴))
84, 5, 73imtr3d 195 . . . 4 (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = 𝐴))
98com12 30 . . 3 (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = 𝐴))
102, 9anim12d 322 . 2 (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴)))
11 df-fn 4932 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12 df-fn 4932 . 2 (tpos 𝐹 Fn 𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴))
1310, 11, 123imtr4g 198 1 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  ccnv 4371  dom cdm 4372  Rel wrel 4377  Fun wfun 4923   Fn wfn 4924  tpos ctpos 5889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937  df-tpos 5890
This theorem is referenced by:  tposfo2  5912  tpos0  5919
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