Theorem hstrlem2 32682

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 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))

Assertion

Ref	Expression
hstrlem2	⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((projℎ‘𝐶)‘𝑢))

Distinct variable groups:   𝑥,𝐶   𝑥,𝑢

Allowed substitution hints:   𝐶(𝑢)   𝑆(𝑥, 𝑢)

Proof of Theorem hstrlem2

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶))
2	1	fveq1d 6887	. 2 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢))
3		hstrlem2.1	. 2 ⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))
4		fvex 6898	. 2 ⊢ ((projℎ‘𝐶)‘𝑢) ∈ V
5	2, 3, 4	fvmpt 6993	1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((projℎ‘𝐶)‘𝑢))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5194  ‘cfv 6540   C_ℋ cch 31352  proj_ℎcpjh 31360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548

This theorem is used by:  hstrlem3a  32683  hstrlem4  32685  hstrlem5  32686