Theorem 2nd1st 8018

Description: Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Assertion

Ref	Expression
2nd1st	⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)

Proof of Theorem 2nd1st

Step	Hyp	Ref	Expression
1		1st2nd2 8008	. . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	sneqd 4633	. . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
3	2	cnveqd 5866	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
4	3	unieqd 4913	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
5		opswap 6219	. 2 ⊢ ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩
6	4, 5	eqtrdi 2780	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {csn 4621  ⟨cop 4627  ∪ cuni 4900   × cxp 5665  ^◡ccnv 5666  ‘cfv 6534  1^st c1st 7967  2^nd c2nd 7968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-1st 7969  df-2nd 7970

This theorem is referenced by:  fcnvgreu  32393  gsumhashmul  32702