Theorem 2nd1st 8020

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 8010	. . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	sneqd 4604	. . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
3	2	cnveqd 5842	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
4	3	unieqd 4887	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
5		opswap 6205	. 2 ⊢ ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩
6	4, 5	eqtrdi 2781	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4592  ⟨cop 4598  ∪ cuni 4874   × cxp 5639  ^◡ccnv 5640  ‘cfv 6514  1^st c1st 7969  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-1st 7971  df-2nd 7972

This theorem is referenced by:  fcnvgreu  32604  gsumhashmul  33008  tposideq  48880  swapf1a  49262  swapf2a  49264