Theorem tposconst 8216

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 7485	. . 3 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	tposmpo 8215	. 2 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
3		fconstmpo 7485	. 2 ⊢ ((𝐵 × 𝐴) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
4	2, 3	eqtr4i 2763	1 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})

Syntax hints:   = wceq 1542  {csn 4582   × cxp 5630   ∈ cmpo 7370  tpos ctpos 8177

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-oprab 7372  df-mpo 7373  df-tpos 8178

This theorem is referenced by:  mattposvs  22411