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Theorem tposfn 8190
Description: Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
tposfn (𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴))

Proof of Theorem tposfn
StepHypRef Expression
1 tposf 8189 . 2 (𝐹:(𝐴 × 𝐵)⟶V → tpos 𝐹:(𝐵 × 𝐴)⟶V)
2 dffn2 6674 . 2 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
3 dffn2 6674 . 2 (tpos 𝐹 Fn (𝐵 × 𝐴) ↔ tpos 𝐹:(𝐵 × 𝐴)⟶V)
41, 2, 33imtr4i 292 1 (𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3447   × cxp 5635   Fn wfn 6495  wf 6496  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-f 6504  df-fo 6506  df-fv 6508  df-tpos 8161
This theorem is referenced by:  tpossym  8193  funcoppc  17769
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