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Theorem opeq 4619
 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.
StepHypRef Expression
1 df-op 4566 . 2 Proj1 A, Proj2 A = ({x y Proj1 Ax = Phi y} ∪ {x y Proj2 Ax = ( Phi y ∪ {0c})})
2 df-proj1 4567 . . . . . . 7 Proj1 A = {z Phi z A}
32rexeqi 2812 . . . . . 6 (y Proj1 Ax = Phi yy {z Phi z A}x = Phi y)
4 phieq 4570 . . . . . . . 8 (z = y Phi z = Phi y)
54eleq1d 2419 . . . . . . 7 (z = y → ( Phi z A Phi y A))
65rexab 2999 . . . . . 6 (y {z Phi z A}x = Phi yy( 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))
98pm5.32i 618 . . . . . . . . 9 ((x = Phi y x A) ↔ (x = Phi y Phi y A))
107, 9bitr4i 243 . . . . . . . 8 (( Phi y A x = Phi y) ↔ (x = Phi y x A))
1110exbii 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))
1411, 12, 133bitri 262 . . . . . 6 (y( Phi y A x = Phi y) ↔ (x A y x = Phi y))
153, 6, 143bitri 262 . . . . 5 (y Proj1 Ax = Phi y ↔ (x A y x = Phi y))
1615abbii 2465 . . . 4 {x y Proj1 Ax = Phi y} = {x (x A y x = Phi y)}
17 df-rab 2623 . . . 4 {x A y x = Phi y} = {x (x A y x = Phi y)}
1816, 17eqtr4i 2376 . . 3 {x y Proj1 Ax = Phi y} = {x A y x = Phi y}
19 df-proj2 4568 . . . . . . 7 Proj2 A = {z ( Phi z ∪ {0c}) A}
2019rexeqi 2812 . . . . . 6 (y Proj2 Ax = ( Phi y ∪ {0c}) ↔ y {z ( Phi z ∪ {0c}) A}x = ( Phi y ∪ {0c}))
214uneq1d 3417 . . . . . . . 8 (z = y → ( Phi z ∪ {0c}) = ( Phi y ∪ {0c}))
2221eleq1d 2419 . . . . . . 7 (z = y → (( Phi z ∪ {0c}) A ↔ ( Phi y ∪ {0c}) A))
2322rexab 2999 . . . . . 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))
2625pm5.32i 618 . . . . . . . . 9 ((x = ( Phi y ∪ {0c}) x A) ↔ (x = ( Phi y ∪ {0c}) ( Phi y ∪ {0c}) A))
2724, 26bitr4i 243 . . . . . . . 8 ((( Phi y ∪ {0c}) A x = ( Phi y ∪ {0c})) ↔ (x = ( Phi y ∪ {0c}) x A))
2827exbii 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})))
3128, 29, 303bitri 262 . . . . . 6 (y(( Phi y ∪ {0c}) A x = ( Phi y ∪ {0c})) ↔ (x A y x = ( Phi y ∪ {0c})))
3220, 23, 313bitri 262 . . . . 5 (y Proj2 Ax = ( Phi y ∪ {0c}) ↔ (x A y x = ( Phi y ∪ {0c})))
3332abbii 2465 . . . 4 {x y Proj2 Ax = ( Phi y ∪ {0c})} = {x (x A y x = ( Phi y ∪ {0c}))}
34 df-rab 2623 . . . 4 {x A y x = ( Phi y ∪ {0c})} = {x (x A y x = ( Phi y ∪ {0c}))}
3533, 34eqtr4i 2376 . . 3 {x y Proj2 Ax = ( Phi y ∪ {0c})} = {x A y x = ( Phi y ∪ {0c})}
3618, 35uneq12i 3416 . 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 3526 . . 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 2788 . . . 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 2862 . . . . . . 7 x V
4039phiall 4618 . . . . . 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})))
4240, 41mpbi 199 . . . . 5 (y x = Phi y y x = ( Phi y ∪ {0c}))
4342a1i 10 . . . 4 (x A → (y x = Phi y y x = ( Phi y ∪ {0c})))
4438, 43mprgbir 2684 . . 3 A = {x A (y x = Phi y y x = ( Phi y ∪ {0c}))}
4537, 44eqtr4i 2376 . 2 ({x A y x = Phi y} ∪ {x A y x = ( Phi y ∪ {0c})}) = A
461, 36, 453eqtrri 2378 1 A = Proj1 A, Proj2 A
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  {crab 2618   ∪ cun 3207  {csn 3737  0cc0c 4374  ⟨cop 4561   Phi cphi 4562   Proj1 cproj1 4563   Proj2 cproj2 4564 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568 This theorem is referenced by:  opeqexb  4620  xpvv  4843  ssrel  4844  proj1eldm  4927  co01  5093  1stfo  5505  2ndfo  5506  swapf1o  5511  otsnelsi3  5805  xpassenlem  6056  xpassen  6057  nncdiv3lem1  6275  dmfrec  6316  fnfreclem2  6318
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