HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hstrlem2 Structured version   Visualization version   GIF version

Theorem hstrlem2 31507
Description: Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
hstrlem2.1 𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))
Assertion
Ref Expression
hstrlem2 (𝐶C → (𝑆𝐶) = ((proj𝐶)‘𝑢))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑢
Allowed substitution hints:   𝐶(𝑢)   𝑆(𝑥,𝑢)

Proof of Theorem hstrlem2
StepHypRef Expression
1 fveq2 6891 . . 3 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6893 . 2 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
3 hstrlem2.1 . 2 𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))
4 fvex 6904 . 2 ((proj𝐶)‘𝑢) ∈ V
52, 3, 4fvmpt 6998 1 (𝐶C → (𝑆𝐶) = ((proj𝐶)‘𝑢))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5231  cfv 6543   C cch 30177  projcpjh 30185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  hstrlem3a  31508  hstrlem4  31510  hstrlem5  31511
  Copyright terms: Public domain W3C validator