Theorem proj2op 4602

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.

Step	Hyp	Ref	Expression
1		df-op 4567	. . . . 5 ⊢ ⟨A, B⟩ = ({x ∣ ∃y ∈ A x = Phi y} ∪ {x ∣ ∃y ∈ B x = ( Phi y ∪ {0c})})
2	1	eleq2i 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 3221	. . . . 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 2863	. . . . . . . . 9 ⊢ z ∈ V
5	4	phiex 4573	. . . . . . . 8 ⊢ Phi z ∈ V
6		snex 4112	. . . . . . . 8 ⊢ {0c} ∈ V
7	5, 6	unex 4107	. . . . . . 7 ⊢ ( Phi z ∪ {0c}) ∈ V
8		eqeq1 2359	. . . . . . . 8 ⊢ (x = ( Phi z ∪ {0c}) → (x = Phi y ↔ ( Phi z ∪ {0c}) = Phi y))
9	8	rexbidv 2636	. . . . . . 7 ⊢ (x = ( Phi z ∪ {0c}) → (∃y ∈ A x = Phi y ↔ ∃y ∈ A ( Phi z ∪ {0c}) = Phi y))
10	7, 9	elab 2986	. . . . . 6 ⊢ (( Phi z ∪ {0c}) ∈ {x ∣ ∃y ∈ A x = Phi y} ↔ ∃y ∈ A ( Phi z ∪ {0c}) = Phi y)
11		phi011 4600	. . . . . . . . 9 ⊢ (z = y ↔ ( Phi z ∪ {0c}) = ( Phi y ∪ {0c}))
12		equcom 1680	. . . . . . . . 9 ⊢ (z = y ↔ y = z)
13	11, 12	bitr3i 242	. . . . . . . 8 ⊢ (( Phi z ∪ {0c}) = ( Phi y ∪ {0c}) ↔ y = z)
14	13	rexbii 2640	. . . . . . 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})))
16	15	rexbidv 2636	. . . . . . . 8 ⊢ (x = ( Phi z ∪ {0c}) → (∃y ∈ B x = ( Phi y ∪ {0c}) ↔ ∃y ∈ B ( Phi z ∪ {0c}) = ( Phi y ∪ {0c})))
17	7, 16	elab 2986	. . . . . . 7 ⊢ (( Phi z ∪ {0c}) ∈ {x ∣ ∃y ∈ B x = ( Phi y ∪ {0c})} ↔ ∃y ∈ B ( Phi z ∪ {0c}) = ( Phi y ∪ {0c}))
18		risset 2662	. . . . . . 7 ⊢ (z ∈ B ↔ ∃y ∈ B y = z)
19	14, 17, 18	3bitr4i 268	. . . . . 6 ⊢ (( Phi z ∪ {0c}) ∈ {x ∣ ∃y ∈ B x = ( Phi y ∪ {0c})} ↔ z ∈ B)
20	10, 19	orbi12i 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))
21	3, 20	bitri 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))
22	2, 21	bitri 240	. . 3 ⊢ (( Phi z ∪ {0c}) ∈ ⟨A, B⟩ ↔ (∃y ∈ A ( Phi z ∪ {0c}) = Phi y ∨ z ∈ B))
23		phieq 4571	. . . . . 6 ⊢ (x = z → Phi x = Phi z)
24	23	uneq1d 3418	. . . . 5 ⊢ (x = z → ( Phi x ∪ {0c}) = ( Phi z ∪ {0c}))
25	24	eleq1d 2419	. . . 4 ⊢ (x = z → (( Phi x ∪ {0c}) ∈ ⟨A, B⟩ ↔ ( Phi z ∪ {0c}) ∈ ⟨A, B⟩))
26		df-proj2 4569	. . . 4 ⊢ Proj2 ⟨A, B⟩ = {x ∣ ( Phi x ∪ {0c}) ∈ ⟨A, B⟩}
27	4, 25, 26	elab2 2989	. . 3 ⊢ (z ∈ Proj2 ⟨A, B⟩ ↔ ( Phi z ∪ {0c}) ∈ ⟨A, B⟩)
28		0cnelphi 4598	. . . . . . . 8 ⊢ ¬ 0c ∈ Phi y
29		ssun2 3428	. . . . . . . . . 10 ⊢ {0c} ⊆ ( Phi z ∪ {0c})
30		0cex 4393	. . . . . . . . . . 11 ⊢ 0c ∈ V
31	30	snid 3761	. . . . . . . . . 10 ⊢ 0c ∈ {0c}
32	29, 31	sselii 3271	. . . . . . . . 9 ⊢ 0c ∈ ( Phi z ∪ {0c})
33		eleq2 2414	. . . . . . . . 9 ⊢ (( Phi z ∪ {0c}) = Phi y → (0c ∈ ( Phi z ∪ {0c}) ↔ 0c ∈ Phi y))
34	32, 33	mpbii 202	. . . . . . . 8 ⊢ (( Phi z ∪ {0c}) = Phi y → 0c ∈ Phi y)
35	28, 34	mto 167	. . . . . . 7 ⊢ ¬ ( Phi z ∪ {0c}) = Phi y
36	35	a1i 10	. . . . . 6 ⊢ (y ∈ A → ¬ ( Phi z ∪ {0c}) = Phi y)
37	36	nrex 2717	. . . . 5 ⊢ ¬ ∃y ∈ A ( Phi z ∪ {0c}) = Phi y
38	37	biorfi 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))
40	38, 39	bitri 240	. . 3 ⊢ (z ∈ B ↔ (∃y ∈ A ( Phi z ∪ {0c}) = Phi y ∨ z ∈ B))
41	22, 27, 40	3bitr4i 268	. 2 ⊢ (z ∈ Proj2 ⟨A, B⟩ ↔ z ∈ B)
42	41	eqriv 2350	1 ⊢ Proj2 ⟨A, B⟩ = B

Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616   ∪ cun 3208  {csn 3738  0_cc0c 4375  ⟨cop 4562   Phi cphi 4563   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-13 1712  ax-14 1714  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-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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  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-pss 3262  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj2 4569

This theorem is referenced by:  opth  4603  opexb  4604