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Theorem tposmpo 8244
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 7410 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
3 ancom 461 . . . . . 6 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
43anbi1i 624 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶))
54oprabbii 7472 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
61, 2, 53eqtri 2764 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
76tposoprab 8243 . 2 tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
8 df-mpo 7410 . 2 (𝑦𝐵, 𝑥𝐴𝐶) = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
97, 8eqtr4i 2763 1 tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  {coprab 7406  cmpo 7407  tpos ctpos 8206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-fv 6548  df-oprab 7409  df-mpo 7410  df-tpos 8207
This theorem is referenced by:  tposconst  8245  oppchomf  17662  oppglsm  19504  mattpos1  21949  mamutpos  21951  madutpos  22135  mdetpmtr2  32792
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