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Theorem hstrlem2 31243
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 6847 . . 3 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6849 . 2 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
3 hstrlem2.1 . 2 𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))
4 fvex 6860 . 2 ((proj𝐶)‘𝑢) ∈ V
52, 3, 4fvmpt 6953 1 (𝐶C → (𝑆𝐶) = ((proj𝐶)‘𝑢))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cmpt 5193  cfv 6501   C cch 29913  projcpjh 29921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509
This theorem is referenced by:  hstrlem3a  31244  hstrlem4  31246  hstrlem5  31247
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