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Theorem tposmpt2 5896
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 5517 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
3 ancom 253 . . . . . 6 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
43anbi1i 431 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶))
54oprabbii 5560 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
61, 2, 53eqtri 2064 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
76tposoprab 5895 . 2 tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
8 df-mpt2 5517 . 2 (𝑦𝐵, 𝑥𝐴𝐶) = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ ((𝑦𝐵𝑥𝐴) ∧ 𝑧 = 𝐶)}
97, 8eqtr4i 2063 1 tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wa 97   = wceq 1243  wcel 1393  {coprab 5513  cmpt2 5514  tpos ctpos 5859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-tpos 5860
This theorem is referenced by: (None)
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