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Theorem proj2op 4601
 Description: The second projection operator applied to an ordered pair yields its second member. Theorem X.2.8 of [Rosser] p. 283. (Contributed by SF, 3-Feb-2015.)
Assertion
Ref Expression
proj2op Proj2 A, B = B

Proof of Theorem proj2op
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-op 4566 . . . . 5 A, B = ({x y A x = Phi y} ∪ {x y B x = ( Phi y ∪ {0c})})
21eleq2i 2417 . . . 4 (( Phi z ∪ {0c}) A, B ↔ ( Phi z ∪ {0c}) ({x y A x = Phi y} ∪ {x y B x = ( Phi y ∪ {0c})}))
3 elun 3220 . . . . 5 (( Phi z ∪ {0c}) ({x y A x = Phi y} ∪ {x y B x = ( Phi y ∪ {0c})}) ↔ (( Phi z ∪ {0c}) {x y A x = Phi y} ( Phi z ∪ {0c}) {x y B x = ( Phi y ∪ {0c})}))
4 vex 2862 . . . . . . . . 9 z V
54phiex 4572 . . . . . . . 8 Phi z V
6 snex 4111 . . . . . . . 8 {0c} V
75, 6unex 4106 . . . . . . 7 ( Phi z ∪ {0c}) V
8 eqeq1 2359 . . . . . . . 8 (x = ( Phi z ∪ {0c}) → (x = Phi y ↔ ( Phi z ∪ {0c}) = Phi y))
98rexbidv 2635 . . . . . . 7 (x = ( Phi z ∪ {0c}) → (y A x = Phi yy A ( Phi z ∪ {0c}) = Phi y))
107, 9elab 2985 . . . . . 6 (( Phi z ∪ {0c}) {x y A x = Phi y} ↔ y A ( Phi z ∪ {0c}) = Phi y)
11 phi011 4599 . . . . . . . . 9 (z = y ↔ ( Phi z ∪ {0c}) = ( Phi y ∪ {0c}))
12 equcom 1680 . . . . . . . . 9 (z = yy = z)
1311, 12bitr3i 242 . . . . . . . 8 (( Phi z ∪ {0c}) = ( Phi y ∪ {0c}) ↔ y = z)
1413rexbii 2639 . . . . . . 7 (y B ( Phi z ∪ {0c}) = ( Phi y ∪ {0c}) ↔ y B y = z)
15 eqeq1 2359 . . . . . . . . 9 (x = ( Phi z ∪ {0c}) → (x = ( Phi y ∪ {0c}) ↔ ( Phi z ∪ {0c}) = ( Phi y ∪ {0c})))
1615rexbidv 2635 . . . . . . . 8 (x = ( Phi z ∪ {0c}) → (y B x = ( Phi y ∪ {0c}) ↔ y B ( Phi z ∪ {0c}) = ( Phi y ∪ {0c})))
177, 16elab 2985 . . . . . . 7 (( Phi z ∪ {0c}) {x y B x = ( Phi y ∪ {0c})} ↔ y B ( Phi z ∪ {0c}) = ( Phi y ∪ {0c}))
18 risset 2661 . . . . . . 7 (z By B y = z)
1914, 17, 183bitr4i 268 . . . . . 6 (( Phi z ∪ {0c}) {x y B x = ( Phi y ∪ {0c})} ↔ z B)
2010, 19orbi12i 507 . . . . 5 ((( Phi z ∪ {0c}) {x y A x = Phi y} ( Phi z ∪ {0c}) {x y B x = ( Phi y ∪ {0c})}) ↔ (y A ( Phi z ∪ {0c}) = Phi y z B))
213, 20bitri 240 . . . 4 (( Phi z ∪ {0c}) ({x y A x = Phi y} ∪ {x y B x = ( Phi y ∪ {0c})}) ↔ (y A ( Phi z ∪ {0c}) = Phi y z B))
222, 21bitri 240 . . 3 (( Phi z ∪ {0c}) A, B ↔ (y A ( Phi z ∪ {0c}) = Phi y z B))
23 phieq 4570 . . . . . 6 (x = z Phi x = Phi z)
2423uneq1d 3417 . . . . 5 (x = z → ( Phi x ∪ {0c}) = ( Phi z ∪ {0c}))
2524eleq1d 2419 . . . 4 (x = z → (( Phi x ∪ {0c}) A, B ↔ ( Phi z ∪ {0c}) A, B))
26 df-proj2 4568 . . . 4 Proj2 A, B = {x ( Phi x ∪ {0c}) A, B}
274, 25, 26elab2 2988 . . 3 (z Proj2 A, B ↔ ( Phi z ∪ {0c}) A, B)
28 0cnelphi 4597 . . . . . . . 8 ¬ 0c Phi y
29 ssun2 3427 . . . . . . . . . 10 {0c} ( Phi z ∪ {0c})
30 0cex 4392 . . . . . . . . . . 11 0c V
3130snid 3760 . . . . . . . . . 10 0c {0c}
3229, 31sselii 3270 . . . . . . . . 9 0c ( Phi z ∪ {0c})
33 eleq2 2414 . . . . . . . . 9 (( Phi z ∪ {0c}) = Phi y → (0c ( Phi z ∪ {0c}) ↔ 0c Phi y))
3432, 33mpbii 202 . . . . . . . 8 (( Phi z ∪ {0c}) = Phi y → 0c Phi y)
3528, 34mto 167 . . . . . . 7 ¬ ( Phi z ∪ {0c}) = Phi y
3635a1i 10 . . . . . 6 (y A → ¬ ( Phi z ∪ {0c}) = Phi y)
3736nrex 2716 . . . . 5 ¬ y A ( Phi z ∪ {0c}) = Phi y
3837biorfi 396 . . . 4 (z B ↔ (z B y A ( Phi z ∪ {0c}) = Phi y))
39 orcom 376 . . . 4 ((z B y A ( Phi z ∪ {0c}) = Phi y) ↔ (y A ( Phi z ∪ {0c}) = Phi y z B))
4038, 39bitri 240 . . 3 (z B ↔ (y A ( Phi z ∪ {0c}) = Phi y z B))
4122, 27, 403bitr4i 268 . 2 (z Proj2 A, Bz B)
4241eqriv 2350 1 Proj2 A, B = B
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615   ∪ cun 3207  {csn 3737  0cc0c 4374  ⟨cop 4561   Phi cphi 4562   Proj2 cproj2 4564 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-13 1712  ax-14 1714  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-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  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-pss 3261  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-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj2 4568 This theorem is referenced by:  opth  4602  opexb  4603
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