Theorem 2nd1st 8017

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 8007	. . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	sneqd 4601	. . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
3	2	cnveqd 5839	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
4	3	unieqd 4884	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
5		opswap 6202	. 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 1540   ∈ wcel 2109  {csn 4589  ⟨cop 4595  ∪ cuni 4871   × cxp 5636  ^◡ccnv 5637  ‘cfv 6511  1^st c1st 7966  2^nd c2nd 7967

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-1st 7968  df-2nd 7969

This theorem is referenced by:  fcnvgreu  32597  gsumhashmul  33001  tposideq  48876  swapf1a  49258  swapf2a  49260