Theorem tposconst 8265

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 7529	. . 3 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	tposmpo 8264	. 2 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
3		fconstmpo 7529	. 2 ⊢ ((𝐵 × 𝐴) × {𝐶}) = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶)
4	2, 3	eqtr4i 2787	1 ⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})

This proof depends on syntax axioms:   = wceq 1570  {csn 4584   × cxp 5649   ∈ cmpo 7414  tpos ctpos 8226

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539  df-oprab 7416  df-mpo 7417  df-tpos 8227

This theorem is used by:  mattposvs  22750