Theorem 2nd1st 7984

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 7974	. . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	sneqd 4580	. . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
3	2	cnveqd 5824	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
4	3	unieqd 4864	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
5		opswap 6187	. 2 ⊢ ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩
6	4, 5	eqtrdi 2788	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4568  ⟨cop 4574  ∪ cuni 4851   × cxp 5622  ^◡ccnv 5623  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7935  df-2nd 7936

This theorem is referenced by:  fcnvgreu  32760  gsumhashmul  33143  tposideq  49375  swapf1a  49756  swapf2a  49758