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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
StepHypRef Expression
1 imakeq2 4225 . 2 (A = B → (kImagek((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A) = (kImagek((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k B))
2 dfproj12 4576 . 2 Proj1 A = (kImagek((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A)
3 dfproj12 4576 . 2 Proj1 B = (kImagek((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k B)
41, 2, 33eqtr4g 2410 1 (A = B Proj1 A = Proj1 B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208   ⊕ csymdif 3209  1cc1c 4134  ℘1cpw1 4135   ×k cxpk 4174  ◡kccnvk 4175   Ins2k cins2k 4176   Ins3k cins3k 4177   “k cimak 4179   SIk csik 4181  Imagekcimagek 4182   Sk cssetk 4183   Ik cidk 4184   Nn cnnc 4373   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-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
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