MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposmpo Structured version   Visualization version   GIF version

Theorem tposmpo 8288
Description: Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
tposmpo.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
tposmpo tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem tposmpo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tposmpo.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 df-mpo 7436 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
3 ancom 460 . . . . . 6 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
43anbi1i 624 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶))
54oprabbii 7500 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
61, 2, 53eqtri 2769 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
76tposoprab 8287 . 2 tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
8 df-mpo 7436 . 2 (𝑦𝐵, 𝑥𝐴𝐶) = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
97, 8eqtr4i 2768 1 tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  {coprab 7432  cmpo 7433  tpos ctpos 8250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-oprab 7435  df-mpo 7436  df-tpos 8251
This theorem is referenced by:  tposconst  8289  oppchomf  17763  oppglsm  19660  mattpos1  22462  mamutpos  22464  madutpos  22648  mdetpmtr2  33823
  Copyright terms: Public domain W3C validator