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Theorem tposfn2 7261
 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 7255 . . . 4 (Fun 𝐹 → Fun tpos 𝐹)
21a1i 11 . . 3 (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3 dmtpos 7251 . . . . . 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 5124 . . . . 5 (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6 cnveq 5218 . . . . . 6 (dom 𝐹 = 𝐴dom 𝐹 = 𝐴)
76eqeq2d 2620 . . . . 5 (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = dom 𝐹 ↔ dom tpos 𝐹 = 𝐴))
84, 5, 73imtr3d 281 . . . 4 (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = 𝐴))
98com12 32 . . 3 (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = 𝐴))
102, 9anim12d 584 . 2 (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴)))
11 df-fn 5807 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12 df-fn 5807 . 2 (tpos 𝐹 Fn 𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴))
1310, 11, 123imtr4g 284 1 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ◡ccnv 5037  dom cdm 5038  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  tpos ctpos 7238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-tpos 7239 This theorem is referenced by:  tposfo2  7262  tpos0  7269
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