Theorem elswap 4741

Description: Membership in the Swap function. (Contributed by SF, 6-Jan-2015.)

Assertion

Ref	Expression
elswap	⊢ (A ∈ Swap ↔ ∃x∃y A = ⟨⟨x, y⟩, ⟨y, x⟩⟩)

Distinct variable group:   x,A,y

Proof of Theorem elswap

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-swap 4725	. . . 4 ⊢ Swap = {⟨z, w⟩ ∣ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)}
2	1	eleq2i 2417	. . 3 ⊢ (A ∈ Swap ↔ A ∈ {⟨z, w⟩ ∣ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)})
3		elopab 4697	. . 3 ⊢ (A ∈ {⟨z, w⟩ ∣ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)} ↔ ∃z∃w(A = ⟨z, w⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)))
4	2, 3	bitri 240	. 2 ⊢ (A ∈ Swap ↔ ∃z∃w(A = ⟨z, w⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)))
5		exrot4 1745	. . 3 ⊢ (∃z∃w∃x∃y(A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ ∃x∃y∃z∃w(A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)))
6		19.42vv 1907	. . . 4 ⊢ (∃x∃y(A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ (A = ⟨z, w⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)))
7	6	2exbii 1583	. . 3 ⊢ (∃z∃w∃x∃y(A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ ∃z∃w(A = ⟨z, w⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)))
8		df-3an 936	. . . . . . 7 ⊢ ((z = ⟨x, y⟩ ∧ w = ⟨y, x⟩ ∧ A = ⟨z, w⟩) ↔ ((z = ⟨x, y⟩ ∧ w = ⟨y, x⟩) ∧ A = ⟨z, w⟩))
9		ancom 437	. . . . . . 7 ⊢ (((z = ⟨x, y⟩ ∧ w = ⟨y, x⟩) ∧ A = ⟨z, w⟩) ↔ (A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)))
10	8, 9	bitr2i 241	. . . . . 6 ⊢ ((A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩ ∧ A = ⟨z, w⟩))
11	10	2exbii 1583	. . . . 5 ⊢ (∃z∃w(A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ ∃z∃w(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩ ∧ A = ⟨z, w⟩))
12		vex 2863	. . . . . . 7 ⊢ x ∈ V
13		vex 2863	. . . . . . 7 ⊢ y ∈ V
14	12, 13	opex 4589	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
15	13, 12	opex 4589	. . . . . 6 ⊢ ⟨y, x⟩ ∈ V
16		opeq1 4579	. . . . . . 7 ⊢ (z = ⟨x, y⟩ → ⟨z, w⟩ = ⟨⟨x, y⟩, w⟩)
17	16	eqeq2d 2364	. . . . . 6 ⊢ (z = ⟨x, y⟩ → (A = ⟨z, w⟩ ↔ A = ⟨⟨x, y⟩, w⟩))
18		opeq2 4580	. . . . . . 7 ⊢ (w = ⟨y, x⟩ → ⟨⟨x, y⟩, w⟩ = ⟨⟨x, y⟩, ⟨y, x⟩⟩)
19	18	eqeq2d 2364	. . . . . 6 ⊢ (w = ⟨y, x⟩ → (A = ⟨⟨x, y⟩, w⟩ ↔ A = ⟨⟨x, y⟩, ⟨y, x⟩⟩))
20	14, 15, 17, 19	ceqsex2v 2897	. . . . 5 ⊢ (∃z∃w(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩ ∧ A = ⟨z, w⟩) ↔ A = ⟨⟨x, y⟩, ⟨y, x⟩⟩)
21	11, 20	bitri 240	. . . 4 ⊢ (∃z∃w(A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ A = ⟨⟨x, y⟩, ⟨y, x⟩⟩)
22	21	2exbii 1583	. . 3 ⊢ (∃x∃y∃z∃w(A = ⟨z, w⟩ ∧ (z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ ∃x∃y A = ⟨⟨x, y⟩, ⟨y, x⟩⟩)
23	5, 7, 22	3bitr3i 266	. 2 ⊢ (∃z∃w(A = ⟨z, w⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ w = ⟨y, x⟩)) ↔ ∃x∃y A = ⟨⟨x, y⟩, ⟨y, x⟩⟩)
24	4, 23	bitri 240	1 ⊢ (A ∈ Swap ↔ ∃x∃y A = ⟨⟨x, y⟩, ⟨y, x⟩⟩)

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4562  {copab 4623   Swap cswap 4719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-swap 4725

This theorem is referenced by:  dfswap2  4742