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Theorem tposmpo 8248
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 7414 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
3 ancom 462 . . . . . 6 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
43anbi1i 625 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶))
54oprabbii 7476 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
61, 2, 53eqtri 2765 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
76tposoprab 8247 . 2 tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
8 df-mpo 7414 . 2 (𝑦𝐵, 𝑥𝐴𝐶) = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
97, 8eqtr4i 2764 1 tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {coprab 7410  cmpo 7411  tpos ctpos 8210
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-oprab 7413  df-mpo 7414  df-tpos 8211
This theorem is referenced by:  tposconst  8249  oppchomf  17666  oppglsm  19510  mattpos1  21958  mamutpos  21960  madutpos  22144  mdetpmtr2  32804
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