Theorem 2nd1st 6206

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 6201	. . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	sneqd 3620	. . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
3	2	cnveqd 4821	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
4	3	unieqd 3835	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩})
5		1stexg 6193	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ V)
6		2ndexg 6194	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ V)
7		opswapg 5133	. . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)
8	5, 6, 7	syl2anc 411	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{⟨(1st ‘𝐴), (2nd ‘𝐴)⟩} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)
9	4, 8	eqtrd 2222	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ⟨(2nd ‘𝐴), (1st ‘𝐴)⟩)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2160  Vcvv 2752  {csn 3607  ⟨cop 3610  ∪ cuni 3824   × cxp 4642  ^◡ccnv 4643  ‘cfv 5235  1^st c1st 6164  2^nd c2nd 6165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243  df-1st 6166  df-2nd 6167

This theorem is referenced by: (None)