Theorem proj1eq 4589

Description: Equality theorem for first projection operator. (Contributed by SF, 2-Jan-2015.)

Assertion

Ref	Expression
proj1eq	|- (A = B -> Proj1 A = Proj1 B)

Proof of Theorem proj1eq

Step	Hyp	Ref	Expression
1		imakeq2 4225	. 2 |- (A = B -> (`'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V)))"kA) = (`'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V)))"kB))
2		dfproj12 4576	. 2 |- Proj1 A = (`'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V)))"kA)
3		dfproj12 4576	. 2 |- Proj1 B = (`'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V)))"kB)
4	1, 2, 3	3eqtr4g 2410	1 |- (A = B -> Proj1 A = Proj1 B)

Syntax hints:   -> wi 4   = wceq 1642  _Vcvv 2859   ∼ ccompl 3205   \ cdif 3206   u. cun 3207   i^i cin 3208   (+) csymdif 3209  1_cc1c 4134  ~P₁ cpw1 4135   X._k cxpk 4174  `'_kccnvk 4175   Ins2_k cins2k 4176   Ins3_k cins3k 4177  "_kcimak 4179   SI_k csik 4181  Image_kcimagek 4182   S_k cssetk 4183   _I_k_k cidk 4184   Nn cnnc 4373   Proj1 cproj1 4563

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 4078  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-v 2861  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-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-phi 4565  df-proj1 4567

This theorem is referenced by:  opth  4602