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Theorem tposfn2 8273
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 8267 . . . 4 (Fun 𝐹 → Fun tpos 𝐹)
21a1i 11 . . 3 (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3 dmtpos 8263 . . . . . 6 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
43a1i 11 . . . . 5 (dom 𝐹 = 𝐴 → (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹))
5 releq 5786 . . . . 5 (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6 cnveq 5884 . . . . . 6 (dom 𝐹 = 𝐴dom 𝐹 = 𝐴)
76eqeq2d 2748 . . . . 5 (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = dom 𝐹 ↔ dom tpos 𝐹 = 𝐴))
84, 5, 73imtr3d 293 . . . 4 (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = 𝐴))
98com12 32 . . 3 (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = 𝐴))
102, 9anim12d 609 . 2 (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴)))
11 df-fn 6564 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12 df-fn 6564 . 2 (tpos 𝐹 Fn 𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴))
1310, 11, 123imtr4g 296 1 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  ccnv 5684  dom cdm 5685  Rel wrel 5690  Fun wfun 6555   Fn wfn 6556  tpos ctpos 8250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-tpos 8251
This theorem is referenced by:  tposfo2  8274  tpos0  8281  tposideq  48788
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