Theorem phieq 4571

Description: Equality law for the Phi operation. (Contributed by SF, 3-Feb-2015.)

Assertion

Ref	Expression
phieq	|- (A = B -> Phi A = Phi B)

Proof of Theorem phieq

Step	Hyp	Ref	Expression
1		imakeq2 4226	. 2 |- (A = B -> (((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) = (((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		dfphi2 4570	. 2 |- Phi A = (((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		dfphi2 4570	. 2 |- Phi B = (((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 -> Phi A = Phi B)

Syntax hints:   -> wi 4   = wceq 1642  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207   u. cun 3208   i^i cin 3209   (+) csymdif 3210  1_cc1c 4135  ~P₁ cpw1 4136   X._k cxpk 4175   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   SI_k csik 4182  Image_kcimagek 4183   S_k cssetk 4184   _I_k_k cidk 4185   Nn cnnc 4374   Phi cphi 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 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-phi 4566

This theorem is referenced by:  phi11  4597  proj1op  4601  proj2op  4602  phialllem1  4617  phialllem2  4618  opeq  4620