Theorem tposconst 8290

Description: The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.)

Assertion

Ref	Expression
tposconst	⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})

Proof of Theorem tposconst

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstmpo 7551	. . 3 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	tposmpo 8289	. 2 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
3		fconstmpo 7551	. 2 ⊢ ((𝐵 × 𝐴) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
4	2, 3	eqtr4i 2767	1 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})

Syntax hints:   = wceq 1539  {csn 4625   × cxp 5682   ∈ cmpo 7434  tpos ctpos 8251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-oprab 7436  df-mpo 7437  df-tpos 8252

This theorem is referenced by:  mattposvs  22462