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Theorem proj1op 4600
 Description: The first projection operator applied to an ordered pair yields its first member. Theorem X.2.7 of [Rosser] p. 282. (Contributed by SF, 3-Feb-2015.)
Assertion
Ref Expression
proj1op Proj1 A, B = A

Proof of Theorem proj1op
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 A, B Phi z ({x y A x = Phi y} ∪ {x y B x = ( Phi y ∪ {0c})}))
3 elun 3220 . . . 4 ( Phi z ({x y A x = Phi y} ∪ {x y B x = ( Phi y ∪ {0c})}) ↔ ( Phi z {x y A x = Phi y} Phi z {x y B x = ( Phi y ∪ {0c})}))
4 vex 2862 . . . . . . 7 z V
54phiex 4572 . . . . . 6 Phi z V
6 eqeq1 2359 . . . . . . . . 9 (x = Phi z → (x = Phi y Phi z = Phi y))
7 phi11 4596 . . . . . . . . . 10 (z = y Phi z = Phi y)
8 equcom 1680 . . . . . . . . . 10 (z = yy = z)
97, 8bitr3i 242 . . . . . . . . 9 ( Phi z = Phi yy = z)
106, 9syl6bb 252 . . . . . . . 8 (x = Phi z → (x = Phi yy = z))
1110rexbidv 2635 . . . . . . 7 (x = Phi z → (y A x = Phi yy A y = z))
12 risset 2661 . . . . . . 7 (z Ay A y = z)
1311, 12syl6bbr 254 . . . . . 6 (x = Phi z → (y A x = Phi yz A))
145, 13elab 2985 . . . . 5 ( Phi z {x y A x = Phi y} ↔ z A)
15 eqeq1 2359 . . . . . . 7 (x = Phi z → (x = ( Phi y ∪ {0c}) ↔ Phi z = ( Phi y ∪ {0c})))
1615rexbidv 2635 . . . . . 6 (x = Phi z → (y B x = ( Phi y ∪ {0c}) ↔ y B Phi z = ( Phi y ∪ {0c})))
175, 16elab 2985 . . . . 5 ( Phi z {x y B x = ( Phi y ∪ {0c})} ↔ y B Phi z = ( Phi y ∪ {0c}))
1814, 17orbi12i 507 . . . 4 (( Phi z {x y A x = Phi y} Phi z {x y B x = ( Phi y ∪ {0c})}) ↔ (z A y B Phi z = ( Phi y ∪ {0c})))
192, 3, 183bitri 262 . . 3 ( Phi z A, B ↔ (z A y B Phi z = ( Phi y ∪ {0c})))
20 phieq 4570 . . . . 5 (x = z Phi x = Phi z)
2120eleq1d 2419 . . . 4 (x = z → ( Phi x A, B Phi z A, B))
22 df-proj1 4567 . . . 4 Proj1 A, B = {x Phi x A, B}
234, 21, 22elab2 2988 . . 3 (z Proj1 A, B Phi z A, B)
24 0cnelphi 4597 . . . . . . 7 ¬ 0c Phi z
25 ssun2 3427 . . . . . . . . 9 {0c} ( Phi y ∪ {0c})
26 0cex 4392 . . . . . . . . . 10 0c V
2726snid 3760 . . . . . . . . 9 0c {0c}
2825, 27sselii 3270 . . . . . . . 8 0c ( Phi y ∪ {0c})
29 eleq2 2414 . . . . . . . 8 ( Phi z = ( Phi y ∪ {0c}) → (0c Phi z ↔ 0c ( Phi y ∪ {0c})))
3028, 29mpbiri 224 . . . . . . 7 ( Phi z = ( Phi y ∪ {0c}) → 0c Phi z)
3124, 30mto 167 . . . . . 6 ¬ Phi z = ( Phi y ∪ {0c})
3231a1i 10 . . . . 5 (y B → ¬ Phi z = ( Phi y ∪ {0c}))
3332nrex 2716 . . . 4 ¬ y B Phi z = ( Phi y ∪ {0c})
3433biorfi 396 . . 3 (z A ↔ (z A y B Phi z = ( Phi y ∪ {0c})))
3519, 23, 343bitr4i 268 . 2 (z Proj1 A, Bz A)
3635eqriv 2350 1 Proj1 A, B = A
 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   Proj1 cproj1 4563 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-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-proj1 4567 This theorem is referenced by:  opth  4602  opexb  4603
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