MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposmpt2 Structured version   Visualization version   GIF version

Theorem tposmpt2 7559
Description: Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
tposmpt2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
tposmpt2 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 tposmpt2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tposmpt2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 df-mpt2 6819 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
3 ancom 465 . . . . . 6 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
43anbi1i 733 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶))
54oprabbii 6876 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
61, 2, 53eqtri 2786 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
76tposoprab 7558 . 2 tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
8 df-mpt2 6819 . 2 (𝑦𝐵, 𝑥𝐴𝐶) = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
97, 8eqtr4i 2785 1 tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  {coprab 6815  cmpt2 6816  tpos ctpos 7521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-oprab 6818  df-mpt2 6819  df-tpos 7522
This theorem is referenced by:  tposconst  7560  oppchomf  16601  oppglsm  18277  mattpos1  20484  mamutpos  20486  madutpos  20670  mdetpmtr2  30220
  Copyright terms: Public domain W3C validator