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Theorem tposfn2 8183
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 8177 . . . 4 (Fun 𝐹 → Fun tpos 𝐹)
21a1i 11 . . 3 (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3 dmtpos 8173 . . . . . 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 5736 . . . . 5 (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6 cnveq 5833 . . . . . 6 (dom 𝐹 = 𝐴dom 𝐹 = 𝐴)
76eqeq2d 2744 . . . . 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 610 . 2 (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴)))
11 df-fn 6503 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12 df-fn 6503 . 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 397   = wceq 1542  ccnv 5636  dom cdm 5637  Rel wrel 5642  Fun wfun 6494   Fn wfn 6495  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-tpos 8161
This theorem is referenced by:  tposfo2  8184  tpos0  8191
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