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Theorem tposfn2 7639
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 7633 . . . 4 (Fun 𝐹 → Fun tpos 𝐹)
21a1i 11 . . 3 (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3 dmtpos 7629 . . . . . 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 5436 . . . . 5 (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6 cnveq 5528 . . . . . 6 (dom 𝐹 = 𝐴dom 𝐹 = 𝐴)
76eqeq2d 2835 . . . . 5 (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = dom 𝐹 ↔ dom tpos 𝐹 = 𝐴))
84, 5, 73imtr3d 285 . . . 4 (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = 𝐴))
98com12 32 . . 3 (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = 𝐴))
102, 9anim12d 604 . 2 (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴)))
11 df-fn 6126 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12 df-fn 6126 . 2 (tpos 𝐹 Fn 𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = 𝐴))
1310, 11, 123imtr4g 288 1 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  ccnv 5341  dom cdm 5342  Rel wrel 5347  Fun wfun 6117   Fn wfn 6118  tpos ctpos 7616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-tpos 7617
This theorem is referenced by:  tposfo2  7640  tpos0  7647
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