Theorem opeq 4620

Description: Any class is equal to an ordered pair. (Contributed by Scott Fenton, 8-Apr-2021.)

Assertion

Ref	Expression
opeq	⊢ A = ⟨ Proj1 A, Proj2 A⟩

Proof of Theorem opeq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 4567	. 2 ⊢ ⟨ Proj1 A, Proj2 A⟩ = ({x ∣ ∃y ∈ Proj1 Ax = Phi y} ∪ {x ∣ ∃y ∈ Proj2 Ax = ( Phi y ∪ {0c})})
2		df-proj1 4568	. . . . . . 7 ⊢ Proj1 A = {z ∣ Phi z ∈ A}
3	2	rexeqi 2813	. . . . . 6 ⊢ (∃y ∈ Proj1 Ax = Phi y ↔ ∃y ∈ {z ∣ Phi z ∈ A}x = Phi y)
4		phieq 4571	. . . . . . . 8 ⊢ (z = y → Phi z = Phi y)
5	4	eleq1d 2419	. . . . . . 7 ⊢ (z = y → ( Phi z ∈ A ↔ Phi y ∈ A))
6	5	rexab 3000	. . . . . 6 ⊢ (∃y ∈ {z ∣ Phi z ∈ A}x = Phi y ↔ ∃y( Phi y ∈ A ∧ x = Phi y))
7		ancom 437	. . . . . . . . 9 ⊢ (( Phi y ∈ A ∧ x = Phi y) ↔ (x = Phi y ∧ Phi y ∈ A))
8		eleq1 2413	. . . . . . . . . 10 ⊢ (x = Phi y → (x ∈ A ↔ Phi y ∈ A))
9	8	pm5.32i 618	. . . . . . . . 9 ⊢ ((x = Phi y ∧ x ∈ A) ↔ (x = Phi y ∧ Phi y ∈ A))
10	7, 9	bitr4i 243	. . . . . . . 8 ⊢ (( Phi y ∈ A ∧ x = Phi y) ↔ (x = Phi y ∧ x ∈ A))
11	10	exbii 1582	. . . . . . 7 ⊢ (∃y( Phi y ∈ A ∧ x = Phi y) ↔ ∃y(x = Phi y ∧ x ∈ A))
12		19.41v 1901	. . . . . . 7 ⊢ (∃y(x = Phi y ∧ x ∈ A) ↔ (∃y x = Phi y ∧ x ∈ A))
13		ancom 437	. . . . . . 7 ⊢ ((∃y x = Phi y ∧ x ∈ A) ↔ (x ∈ A ∧ ∃y x = Phi y))
14	11, 12, 13	3bitri 262	. . . . . 6 ⊢ (∃y( Phi y ∈ A ∧ x = Phi y) ↔ (x ∈ A ∧ ∃y x = Phi y))
15	3, 6, 14	3bitri 262	. . . . 5 ⊢ (∃y ∈ Proj1 Ax = Phi y ↔ (x ∈ A ∧ ∃y x = Phi y))
16	15	abbii 2466	. . . 4 ⊢ {x ∣ ∃y ∈ Proj1 Ax = Phi y} = {x ∣ (x ∈ A ∧ ∃y x = Phi y)}
17		df-rab 2624	. . . 4 ⊢ {x ∈ A ∣ ∃y x = Phi y} = {x ∣ (x ∈ A ∧ ∃y x = Phi y)}
18	16, 17	eqtr4i 2376	. . 3 ⊢ {x ∣ ∃y ∈ Proj1 Ax = Phi y} = {x ∈ A ∣ ∃y x = Phi y}
19		df-proj2 4569	. . . . . . 7 ⊢ Proj2 A = {z ∣ ( Phi z ∪ {0c}) ∈ A}
20	19	rexeqi 2813	. . . . . 6 ⊢ (∃y ∈ Proj2 Ax = ( Phi y ∪ {0c}) ↔ ∃y ∈ {z ∣ ( Phi z ∪ {0c}) ∈ A}x = ( Phi y ∪ {0c}))
21	4	uneq1d 3418	. . . . . . . 8 ⊢ (z = y → ( Phi z ∪ {0c}) = ( Phi y ∪ {0c}))
22	21	eleq1d 2419	. . . . . . 7 ⊢ (z = y → (( Phi z ∪ {0c}) ∈ A ↔ ( Phi y ∪ {0c}) ∈ A))
23	22	rexab 3000	. . . . . 6 ⊢ (∃y ∈ {z ∣ ( Phi z ∪ {0c}) ∈ A}x = ( Phi y ∪ {0c}) ↔ ∃y(( Phi y ∪ {0c}) ∈ A ∧ x = ( Phi y ∪ {0c})))
24		ancom 437	. . . . . . . . 9 ⊢ ((( Phi y ∪ {0c}) ∈ A ∧ x = ( Phi y ∪ {0c})) ↔ (x = ( Phi y ∪ {0c}) ∧ ( Phi y ∪ {0c}) ∈ A))
25		eleq1 2413	. . . . . . . . . 10 ⊢ (x = ( Phi y ∪ {0c}) → (x ∈ A ↔ ( Phi y ∪ {0c}) ∈ A))
26	25	pm5.32i 618	. . . . . . . . 9 ⊢ ((x = ( Phi y ∪ {0c}) ∧ x ∈ A) ↔ (x = ( Phi y ∪ {0c}) ∧ ( Phi y ∪ {0c}) ∈ A))
27	24, 26	bitr4i 243	. . . . . . . 8 ⊢ ((( Phi y ∪ {0c}) ∈ A ∧ x = ( Phi y ∪ {0c})) ↔ (x = ( Phi y ∪ {0c}) ∧ x ∈ A))
28	27	exbii 1582	. . . . . . 7 ⊢ (∃y(( Phi y ∪ {0c}) ∈ A ∧ x = ( Phi y ∪ {0c})) ↔ ∃y(x = ( Phi y ∪ {0c}) ∧ x ∈ A))
29		19.41v 1901	. . . . . . 7 ⊢ (∃y(x = ( Phi y ∪ {0c}) ∧ x ∈ A) ↔ (∃y x = ( Phi y ∪ {0c}) ∧ x ∈ A))
30		ancom 437	. . . . . . 7 ⊢ ((∃y x = ( Phi y ∪ {0c}) ∧ x ∈ A) ↔ (x ∈ A ∧ ∃y x = ( Phi y ∪ {0c})))
31	28, 29, 30	3bitri 262	. . . . . 6 ⊢ (∃y(( Phi y ∪ {0c}) ∈ A ∧ x = ( Phi y ∪ {0c})) ↔ (x ∈ A ∧ ∃y x = ( Phi y ∪ {0c})))
32	20, 23, 31	3bitri 262	. . . . 5 ⊢ (∃y ∈ Proj2 Ax = ( Phi y ∪ {0c}) ↔ (x ∈ A ∧ ∃y x = ( Phi y ∪ {0c})))
33	32	abbii 2466	. . . 4 ⊢ {x ∣ ∃y ∈ Proj2 Ax = ( Phi y ∪ {0c})} = {x ∣ (x ∈ A ∧ ∃y x = ( Phi y ∪ {0c}))}
34		df-rab 2624	. . . 4 ⊢ {x ∈ A ∣ ∃y x = ( Phi y ∪ {0c})} = {x ∣ (x ∈ A ∧ ∃y x = ( Phi y ∪ {0c}))}
35	33, 34	eqtr4i 2376	. . 3 ⊢ {x ∣ ∃y ∈ Proj2 Ax = ( Phi y ∪ {0c})} = {x ∈ A ∣ ∃y x = ( Phi y ∪ {0c})}
36	18, 35	uneq12i 3417	. 2 ⊢ ({x ∣ ∃y ∈ Proj1 Ax = Phi y} ∪ {x ∣ ∃y ∈ Proj2 Ax = ( Phi y ∪ {0c})}) = ({x ∈ A ∣ ∃y x = Phi y} ∪ {x ∈ A ∣ ∃y x = ( Phi y ∪ {0c})})
37		unrab 3527	. . 3 ⊢ ({x ∈ A ∣ ∃y x = Phi y} ∪ {x ∈ A ∣ ∃y x = ( Phi y ∪ {0c})}) = {x ∈ A ∣ (∃y x = Phi y ∨ ∃y x = ( Phi y ∪ {0c}))}
38		rabid2 2789	. . . 4 ⊢ (A = {x ∈ A ∣ (∃y x = Phi y ∨ ∃y x = ( Phi y ∪ {0c}))} ↔ ∀x ∈ A (∃y x = Phi y ∨ ∃y x = ( Phi y ∪ {0c})))
39		vex 2863	. . . . . . 7 ⊢ x ∈ V
40	39	phiall 4619	. . . . . 6 ⊢ ∃y(x = Phi y ∨ x = ( Phi y ∪ {0c}))
41		19.43 1605	. . . . . 6 ⊢ (∃y(x = Phi y ∨ x = ( Phi y ∪ {0c})) ↔ (∃y x = Phi y ∨ ∃y x = ( Phi y ∪ {0c})))
42	40, 41	mpbi 199	. . . . 5 ⊢ (∃y x = Phi y ∨ ∃y x = ( Phi y ∪ {0c}))
43	42	a1i 10	. . . 4 ⊢ (x ∈ A → (∃y x = Phi y ∨ ∃y x = ( Phi y ∪ {0c})))
44	38, 43	mprgbir 2685	. . 3 ⊢ A = {x ∈ A ∣ (∃y x = Phi y ∨ ∃y x = ( Phi y ∪ {0c}))}
45	37, 44	eqtr4i 2376	. 2 ⊢ ({x ∈ A ∣ ∃y x = Phi y} ∪ {x ∈ A ∣ ∃y x = ( Phi y ∪ {0c})}) = A
46	1, 36, 45	3eqtrri 2378	1 ⊢ A = ⟨ Proj1 A, Proj2 A⟩

Syntax hints:   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  {crab 2619   ∪ cun 3208  {csn 3738  0_cc0c 4375  ⟨cop 4562   Phi cphi 4563   Proj1 cproj1 4564   Proj2 cproj2 4565

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-rab 2624  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-0c 4378  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  opeqexb  4621  xpvv  4844  ssrel  4845  proj1eldm  4928  co01  5094  1stfo  5506  2ndfo  5507  swapf1o  5512  otsnelsi3  5806  xpassenlem  6057  xpassen  6058  nncdiv3lem1  6276  dmfrec  6317  fnfreclem2  6319