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Theorem tposf 7916
Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposf (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)

Proof of Theorem tposf
StepHypRef Expression
1 relxp 5572 . . 3 Rel (𝐴 × 𝐵)
2 tposf2 7912 . . 3 (Rel (𝐴 × 𝐵) → (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐴 × 𝐵)⟶𝐶))
31, 2ax-mp 5 . 2 (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐴 × 𝐵)⟶𝐶)
4 cnvxp 6013 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
54feq2i 6505 . 2 (tpos 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)
63, 5sylib 219 1 (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   × cxp 5552  ccnv 5553  Rel wrel 5559  wf 6350  tpos ctpos 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-tpos 7888
This theorem is referenced by:  tposfn  7917  mattposcl  20997  tposmap  21001
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