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Theorem hstrlem2 31439
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 6879 . . 3 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6881 . 2 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
3 hstrlem2.1 . 2 𝑆 = (𝑥C ↦ ((proj𝑥)‘𝑢))
4 fvex 6892 . 2 ((proj𝐶)‘𝑢) ∈ V
52, 3, 4fvmpt 6985 1 (𝐶C → (𝑆𝐶) = ((proj𝐶)‘𝑢))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5225  cfv 6533   C cch 30109  projcpjh 30117
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 5293  ax-nul 5300  ax-pr 5421
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 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541
This theorem is referenced by:  hstrlem3a  31440  hstrlem4  31442  hstrlem5  31443
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