Theorem tposconst 7941

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 7264	. . 3 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	tposmpo 7940	. 2 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
3		fconstmpo 7264	. 2 ⊢ ((𝐵 × 𝐴) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
4	2, 3	eqtr4i 2785	1 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})

Syntax hints:   = wceq 1539  {csn 4523   × cxp 5523   ∈ cmpo 7153  tpos ctpos 7902

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-oprab 7155  df-mpo 7156  df-tpos 7903

This theorem is referenced by:  mattposvs  21156